Topology beyond application: Drawing social and mathematical worlds into rhythm

Abstract

For more than sixty years, topology has provided geographers with tools for lifting the veils of Euclideanism and showing how spaces are relational, agential, embodied, and ontologically multiple. A field of mathematics since the nineteenth century, topology has been reworked and mobilised by geographers in diverse ways: from spatial science, through networks and assemblages, to the influences of Law and Mol, Deleuze and Guattari, Agamben, Barad, and others. Recently, geographers have called for employing topology with greater critical discernment, while continuing to develop its promising and subversive potential.

In this article, we aim to help enrich topology's presence in geography and the social sciences more broadly. Following a brief historical introduction, we examine the bodily basis of mathematical reasoning in order to undo the qualitative/quantitative binary and outline a creative, affective approach to abstraction. This lays the ground for the main contribution of the article, where we detail, with several diagrams, key concepts in mathematical topology which critically engage and augment geographers’ theories of space. The potential for an unlikely alliance emerges between geography and mathematics, with implications for both disciplines – provided the inseparability of object and subject, and of knowledge and social order, are carefully attended to.

Introduction

If you go towards it with an open heart and a real, watchful reverence … you will be absolutely amazed at what it will reveal to you.

– John O’Donohue (2008) on landscape

Living in modernity, it can be far from obvious that Euclidean space and linear time are not absolute truths of the world, but simply one model of its many rhythms (Abram, 1997; Deloria Jr, 1979; Massey, 2005). Ticking clocks, Google calendar schedules, and grid-like urban planning all seem to confirm the ordering of Newton's clockwork universe and Descartes’ three dimensions of res extensa. Kwakwa̱ka̱ʼwakw artist and scholar Nicolson (2013), however, describes a very different sense of the world, one which traces closely the contours of both the Kwak’wala language and the lands and waters of her ancestral territories. Space and time become apparent in terms of cycles (ecological, celestial), alternations (light and dark, physical and metaphysical), and expansions and contractions (between body, house, and land).

Nicolson, along with many First Nations scholars (e.g., Atleo, 2007; Hunt, 2025; Long, 2024; Watts, 2013), speaks of a reality for which the rigidity and scalable uniformity of Euclideanism – imposed and sustained by European colonisation (Nicolson, 2013: 378–380) – is inadequate for making sense of, and living with, the more-than-human world. Instead, as Kanien'kehá:ka mathematician Doolittle (2018) suggests, one might look to other fields of mathematics for spatial languages that have greater potential for attuning to lived experience and multiple ways of knowing. Among these is topology, which – as evidenced through this article – shifts focus from linearity and metric distance to notions including relationality, openness and closedness, continuity and discontinuity, and invariance within transformation.

A brief history

A field of mathematics since the nineteenth century (James 1999), topology has a patchy but enduring history in the social sciences (Martin and Secor, 2014). According to Merriman (2022: 77–82), early deployments of topology in geography came in the 1950s–70s under the auspices of a positivist spatial science. It provided a means for modelling flows of movement mathematically, such as the ‘time-space convergence’ induced by transportation and migration. However, this approach was increasingly critiqued – not least by scholars such as Gunnar Olsson and David Harvey, who had previously figured in the development of spatial science – as depoliticising space, and effacing human difference in favour of generalised statements about populations of rational economic actors (Merriman, 2022: 83–88). Nonetheless, this period helped move human geography beyond regional thinking and implicated spatial relations as a focus of the discipline, and the practice of geography as a spatial science continues (Cresswell, 2024: 98).

By the 1990s, topology had in many settings become synonymous with ‘networks’. These were used, for example, in studies of information and commodity travel through globalisation (Castells, 1996), and the spread of scientific knowledge through ‘immutable mobiles’ (Latour, 1987) in what later became known as actor-network theory. A key idea with networks was to study spatial configurations of actors through their social, material, or economic relations, rather than cartographic distances between them. This afforded an important step beyond Euclidean spatial assumptions, particularly in actor-network theory, which evidenced how categories such as ‘space’ or ‘time’ – and binaries such as ‘macro’ and ‘micro’, or ‘natural’ and ‘social’ – are effects (or outcomes) of relations maintained by continually unfolding practices, rather than essential, intrinsic orderings of reality.

Yet in both geography and mathematics, networks present just a sliver of topology. Our article builds from the post-network spatialities emerging in the social sciences around the turn of the century, many of which developed in response to the increasingly apparent limitations of network thinking.

In anthropology, Strathern (1996) illustrates the tendency for Euro-American network ideas to import a priori categories such as human and nonhuman, or hierarchical ideas of relatedness. Instead, she employs fractals as a means for rethinking cross-cultural comparison, questions of complexity and scale in particular (Strathern, 1991). Ingold (2011: 63–94) critiques simplistic renderings of networks for directing attention away from where life is happening: connecting nodes with links might be useful for mapping a fixed assemblage of things, but they are not lines ‘along which anything moves or grows’. Drawing on the rhizomes and lines of becoming of Deleuze and Guattari (1987), Ingold turns networks inside out with the topology of a ‘meshwork’ (borrowing the term from Lefebvre, 1991). He takes bundles of lines as open-ended and improvisational currents along which life is lived, and nodes as moments of lives joining and connecting – an approach continued in his later work on correspondence (Ingold, 2017).

Partially inspired by Strathern's use of fractals and other feminist critiques of networks (e.g., Star, 1990), the spatialities described by Law, Mol, and their collaborators remain among the most generative and innovative topological works in the social sciences (see de Laet and Mol, 2000; Law, 1999, 2002; Law and Mol, 2001; Law and Singleton, 2005; Mol and Law, 1994). Akin to Strathern and Ingold, Mol and Law arrived at topology through the limitations of Euclidean and network thinking, their focus being the co-constitution of objects and spaces.

As Law (2002: 91) details, the material-semiotic approach of actor-network theory proposed that an ‘object’ is an effect of a stable network of relations – so long as these relations don’t change, the object remains, as he illustrates with an example of Portuguese maritime imperialism. The issue, though, is that network objects implicitly produce an encompassing network space, which, like Euclidean space, cannot see its own boundaries and ends up colonising other spatialities into its own image (Law, 2002: 96–98). The success of actor-network theory displaced the hegemonic position of Euclidean space, only to reinstate it with networks (Law, 1999). Mol and Law, however, are interested in how to think well about difference, otherness, and nonconformity. ¹ In a series of seminal case studies concerning anaemia, bush pumps, and liver disease, they collaboratively detail alternative topologies which resist unitary orderings of Euclideanism or networks – most significantly, fluid space, to which we return in the concluding section of this article.

Recent topologies in geography

These and other topologies have been employed to diverse ends in human geography, most prominently under the rubric of relationality (Cresswell, 2024: 212–229; Lury et al., 2012), often with an accompanying opposition to topography and Euclidean space (e.g., Allen, 2011; see also Barad, 2007). This is evident throughout a Theory, Culture & Society special issue: ‘[t]opology sets aside the privilege granted to Euclidean space … topology offers the insight that it is possible to analyze systematically relationships and configurations themselves’ (Shields, 2012: 48; 51).

But in its presentation as a master language for relationality, topology is chained even as it is crowned – there is an echo here of how networks lost their subtleties and subversive pluralities as they became a dominant spatial language. Strathern (2020) details how the term ‘relation’ in Anglophone usage ² carries a positive, benign gloss – to show something is relational is usually assumed to be a good thing. These connotations of connectivity and similarity can make it difficult to notice relations of divergence, multiplicity, incommensurability, ³ or what Gregory (2004) calls ‘connective dissonance’. This is in no small part why geographers look to work with topology beyond a general appeal to relationality (Jones, 2009; Lata and Minca, 2016).

As Martin and Secor (2014: 426) note: ‘[t]opology's field of deployment in geography is not cut from a single fabric’. Focusing on multiplicity and intersecting spatialities, Hinchliffe et al. (2013, 2017) describe how borders between a sanitised inside and contaminated outside (which could be called regional, territorial, or Euclidean thinking), and pathogen movement extending across space from hub to hub (network thinking), are empirically untenable for understanding how diseases travel, and inadequate for managing their spread. The authors work with topology to understand disease outbreaks through spatial intension rather than extension – the result of crossing a threshold of intensity, or density, in fluidly defined borderlands, rather than moving across neatly demarcated borderlines. Drawing on this and earlier topological ideas by Whatmore (2002), Lorimer (2015: 166–167) employs the fluid space of Law and Mol to show how binary regional thinking (and its implicit social ordering) common in ‘fortress conservation’ may be blind to the flourishing of novel urban ecosystems.

These and other empirical studies evidence topology's counter-hegemonic potential, especially for bringing to light otherwise hidden power relations, for which Allen (2011, 2016) has been influential. In the Palestinian governorate of Ramallah and al-Bireh, Harker (2017) traces how the binds of debt – usually considered a temporal relation, buying future labour with present consumption – emerge, and are held in place, through a multitude of spatial relations. Here, topology is used to highlight how statistical measures of debt, and topographical framings of Israeli settler-colonial occupation, underplay the ‘debt ecologies’ which dissolve distances between familial, state, and financial actors to affect the lives of Ramallah's citizens.

Resonant articulations of power operating in a diffuse manner are given by Mittal (2025) in his study on ‘smart cities’ in India, where topology provides a language for understanding how urban governance and spatial planning are shaped by, and contingent on, ephemeral interactions between federal, city, corporate, and event industry representatives.

Many more instances of geographical topology could be noted. Some, such as the work of Agamben (1998) and its uptake in human geography (notably through Gregory, 2004), appear later in this article. For the influences of Deleuze, Massey, Thrift, Serres, and others, and for details of how topology's trajectories in geography intertwine with poststructuralism and nonrepresentational theory, see Martin and Secor (2014) and Merriman (2022).

Topology, then, is a promiscuous and polyphonic character, entangled with significant theoretical junctures and a number of seminal approaches to space in the social sciences, geography in particular. It may be associated with fluid objects, power, practices, networks, globalisation, relationality, psychoanalysis, smart cities, poststructuralism, fractals, pathogens, or wildlife conservation, to name a handful. But for all its proliferation and enduring allure, the conceptual repository deployed by geographers in topological arguments remains relatively generic and without sustained development (a matter not helped by mathematicians making their discipline so inaccessible, to their detriment). Moreover, mathematical topology is often taken as a fixed body of knowledge, missing critical analysis of either the modernist, Euro-American baggage such ideas may import (Law, 2004), or the potential of mathematical topology itself to be fruitfully reworked through geographers’ theories of space.

Paper overview

In this article, our primary aim is to enrich topology's presence in geography and in the social sciences more generally.

We agree with Martin and Secor (2014) that topology's potential in geography closely relates to how it is reworked through poststructuralist theories of space. This is challenged by a number of tensions which, without careful attention, can preclude a generative, nonhierarchical exchange between geography and mathematics. Section 2 therefore works to set up the paper by drawing these disciplines into rhythm – that is, eliciting resonance without eliding difference – in two steps. First, a brief review of social and cognitive studies on the bodily basis of mathematical reasoning makes the case that qualitative/quantitative and metaphorical/mathematical are not dichotomies: rather, in each pair, the latter relies upon (and is never without) the former. The second step builds on this to provide an affirmative critique of abstract approaches to space, detailing how certain modes of abstraction – which we term ‘attunement’, inspired by nehiyaw scholar Long (2024) – can serve to heighten the senses and lure attention beyond taken-for-granted ideas of space. This will inform our sense of topology's ontological significance, not because of any claims that the world ‘is’ topological but because topology affords exploration of what worlds – across both subjects and objects of knowledge – are being made real or realised. ⁴ We will aim to not just describe realities but also to intervene and participate in them, for ‘[t]he problem with ontology is not knowledge or representation, but engagement with and for a world’ (Stengers, 2018a, 85) – it can have very direct legal and political implications (see Long, 2024: 15–18).

The above lays requisite ground for Section 3, which seeks to make more explicit the ideas at work in mathematical topology and connect these with geographers’ uses of topology. It begins by detailing conceptual foundations underpinning ideas of a ‘topological space’ in mathematics. The following three subsections, each building on what precedes, take this in slightly different directions and speak to key debates on topology in geography. The first aims to show how Euclidean space, topologically considered, is already relationally enacted and contingent on human subjectivity – our approach thus departs from the aforementioned presentations of topology as antithetical to topography and Euclidean space. This argument is then drawn into dialogue with Hinchliffe et al. (2013; 2017). Second, a method is described for folding and suturing fragments of Euclidean space to construct objects which may dislodge modernist spatial thinking. We use this to suggest a spatial metaphor attuned to the spaces of exception described by Agamben (1998), Gregory (2004), and others. The third describes a notion of ‘homeomorphism’ from mathematical topology to engage critically with discussions of change and continuity in the social sciences, highlighting how these often retain implicit assumptions of a background space and time, whence we motivate an augmented approach to invariance.

The implications of Section 3 are strange, for topology turns out to cut both ways. Mathematics, when drawn into rhythm with geographical topology, emerges as far more mutable, multiple, and allied to spatial insights in the social sciences than is usually apparent. Indeed, mathematical knowledge itself is evidenced as relationally constituted, its visible faces contingent on the subjectivities embroiled within it. Reciprocal prospects therefore arise: while geographers might stand to gain from further engagement with the richness of mathematical topology to nuance and develop their own conceptual vocabularies, mathematicians in turn may be critiqued for missing the potential of topology to reconfigure subjects as well as objects of knowledge (a potential observed by Secor, 2013; Stirling, 2019). In conclusion, Section 4 takes up this matter by addressing the idea of topology as something to be ‘applied’ to ‘the real world’ or to other fields of study. Through attending to the inseparability of knowledge and social order, and the entanglement of subject and object – drawing in particular on the fluid space of Law and Mol – we advocate working with topology beyond any unitary authority of application.

The value and meaning of mathematical topology for the reader should not be taken as a given, but rather measured against its relevance to one's matters of concern. The ends to which mathematicians deploy topology need not dictate its use in geography – as with any craft, what's important is not just the tools you have but also how you use them, and tool makers need not be the most accomplished tool users. Hence, we aim to write with fidelity not to the symbolic language of mathematical topology, but to the conceptual work it does and ways of thinking it supports. Being a transposition of register, a changing of tongues, the following account may therefore be difficult for many mathematicians to recognise, since we will be invoking geographers’ vocabularies of space. But as mathematical topologist Thurston (1994) writes, although adherence to formal proof and logical syntax has a central place in mathematical exposition, it is also important to express mathematics in ways that engender conceptual understanding, as well as play and experimentation.

Writing about spatial ideas, we wish also to enact them; rather than pinning down and defining topology, we endeavour to entice its many faces into presence. This is germane to our varied interests. Siddharth is a mathematician and geographer, working across disciplines to collaboratively develop mathematics as an ecological practice (Unnithan Kumar, 2024). Andy is a transdisciplinary sociologist engaging with power in science and technology policy. David is a cultural ecologist, geophilosopher, and sleight-of-hand magician. Sanjiv is a retired mathematician and astronomer. Brought together by social relations and a shared fascination in topology, we did not a priori conceive the contents of this article and then inscribe them onto a tabula rasa. Quite the opposite: we resonate with Stengers (2012) in that the contents of this article emerged from the conceptual friction generated by the transformative acts of conversing, listening, writing, and diagramming themselves.

Before continuing, there is something to name. In what has been said and what is to come, reference is made to a number of empirical contexts. That we are even able to consider these as ‘empirical contexts’ indicates our experiential distance, to differing degrees among the present authors, from the lived, political realities of these places – a point made by Black, Indigenous, and feminist scholars, among others (Daley and Murrey, 2022; Haraway, 1988; Long, 2024; Sharpe, 2016; TallBear, 2014; Todd, 2016). We invite the reader to be aware of this (if not already evident), and to pause and make space for their corporeally felt, affective experience when reading this article.

Coming into rhythm

Something spoken has an effect in multiple other domains. It resonates.

– Walter Lightning (1992: 21), quoted in Keara Long (2024: 42)

Beyond qualitative and quantitative: the bodily, metaphorical basis of mathematical reasoning

Given the presently widespread authority and privilege granted to quantification, especially in the sciences, one might think that the emphasis on qualitative and visual reasoning in topology should make it a subversive field of Euro-American mathematics. Indeed, topology can supposedly accommodate qualitative phenomena without the ‘intolerant view of a dogmatically quantitative science’ (Thom, 2018 [1972] [1972]: 6), and promises a ‘number-free and metric-free account of experience’ (see also Prager and Reiners, 2009; Sha, 2012: 242). But is it sufficient to predicate topology's appeal in terms of qualitative versus quantitative?

In their study on mathematics and the ‘embodied mind’, Lakoff and Núñez (2000) draw on empirical research into mathematical reasoning in cognitive science, together with the theory of conceptual metaphor from cognitive linguistics, to understand why mathematical results hold as seemingly solid and unitary truths – not on the levels of deductive logic and formal proof, but on that of ideas, concepts, and meanings. One can think with the insights of their work without subscribing to the cognitivist paradigm and its modernist assumptions about space and materiality (we return to this issue in Section 2.2, ‘Modes of abstraction’).

Lakoff and Núñez demonstrate a continuity between mathematical and everyday concepts, whereby the former are shown to be specific instances and refinements of the latter. For example, the mathematical ideas of being ‘in’ and ‘out’ of a set are conventionally conceptualised and visualised in terms of the concepts of ‘inside’ and ‘outside’ arising from everyday experience with objects and containers (Lakoff and Núñez, 2000: 43–45). The standard rules of arithmetic (e.g., 1 + 2 = 2 + 1) are effective for counting physical objects precisely because these laws may be conceptualised in terms of lived experience handling physical objects (Lakoff and Núñez, 2000: 95–96). ⁵ In particular, the authors emphasise that ‘[m]etaphor is not a mere embellishment; it is the basic means by which abstract thought is made possible’ (Lakoff and Núñez, 2000: 39).

Ernest (2021: 48) writes similarly: ‘Only when we have conferred objecthood on a sequence or grouping of newly recognised entities are they available for counting…. Counting is not a first act. It must always be secondary to our act of parcelling some part of the world into countables’. Furthermore, despite Leopold Kronecker's claim that ‘God made the integers, all else is the work of man’, ⁶ how the world is parcelled into units for counting is not a human universal. For example, according to Ascher (1991: 9), the Yuki may count with a numerical base of eight, rather than ten, by enumerating the spaces between fingers rather than the fingers themselves. ⁷ Or consider Strathern's (2021) account of how the bodily counting practices of the Maenge, Murik, and Hagen, among other Melanesian societies, are immanent to (and also influence) the ways in which the regeneration of both human persons and cultivated plants are understood – so that ‘one’ is less a stable and unified entity than ‘two’, and moreover numbers need not be static and ‘divorced from life's unfolding’. In short, quantity may distil quality, but is never without it – and that too in multiple and equally valid ways (Kwan and Schwanen, 2009; Stirling, 2023).

Now, according to Ernest (2023: 8), the number 1 is often first learned as the beginning word of a spoken count or written tally. Its early use is therefore enactive, inseparable from the process of counting things. With reference to the work of Butler (1990) on gender performativity, Ernest (2023: 11) describes how the coalescing of ‘1 as process’ to ‘1 as object’ – known as reification, envisioning the result of processes as entities in their own right – means that ‘mathematical terms create, over time, the objects to which they refer’. However, the object-oriented language of Anglophone mathematics can conceal these metaphorical and processual foundations of mathematical entities (Sfard, 1994: 47). ⁸

Drawing on interviews with three well-known mathematicians, Sfard (1994: 48) describes how this reification of process into object provides the spatial metaphors on which mathematical insight rests. One interviewee remarks:

To understand a new concept I must create an appropriate metaphor. A personification. Or a spatial metaphor. A metaphor of structure. Only then can I answer questions, solve problems.

It is when a spatial metaphor reifies to become a felt, haptic presence, that certain concepts and terms used in mathematical topology, like ‘surgery’ and ‘excision’, begin to make sense – literally, that is, they begin to speak to the senses (cf. Abram, 1997: 265). Mathematics, in more ways than one, is a full-bodied endeavour.

Spatial metaphors also support the cornerstone of mathematical reasoning that is deductive logic. According to Lakoff and Núñez (2000: 43): ‘much of what is often called logical inference is in fact spatial inference mapped onto an abstract logical domain’. Moreover, deductive reasoning, which may seem linear and deterministic, in practice plays a generative and improvisational role in mathematical inquiry, precisely because mathematics is a bodily skill not separable from movement and gesture. For example, as the hand writes the implication symbol ⇒, the thinking process is carried along in the momentum of the argument's becoming. ⁹ Hence, formal logical deduction is not, even in its abstract mathematical form, separate from spatiality and metaphor (Sinclair, 2023).

This is a very brief signpost to social and cognitive perspectives on spatiality, performativity, and embodiment in mathematics. We draw upon these to make the following point: despite differences between geography and mathematics as epistemic communities, there is no absolute distinction in their uses of topology that can be drawn along a dichotomous line bifurcating metaphorical and mathematical, qualitative and quantitative, or spatial and logical. In each case, the former engender and support the latter through their provision of meaning and coherence.

Modes of abstraction

Metaphor and abstraction are in close relationship. De Freitas and Sinclair, drawing on Barad, Deleuze, and Châtelet, challenge cognitivist approaches to embodiment in mathematics (e.g., Lakoff and Núñez, 2000): ‘treating all concepts as metaphorical in relation to the “real” reinforces the divide between the mathematically abstract and the physically concrete’, implicating the body as a ‘mute, precognitive container for consciousness’ (de Freitas and Sinclair, 2014: 200; 40). Indeed, appealing to metaphor as the ‘mechanism’ for translating bodily experience to abstract concepts implies a mechanical substratum to mathematical reasoning – without appreciating that mechanisms are also metaphors (Abram, 1991).

Due to its deterministic rendering of linear cause and effect (Barad, 2007) – and the alienating, dehumanising, and patriarchal ends towards which it has been used (Blackie, 2016; Federici, 2004; Merchant, 1980) – we have no wish to further the mechanical worldview of what Whitehead called ‘scientific materialism’ (Roberts, 2024). While mechanistic logics often seem to follow applications of mathematics as sure as tracks follow the wheels of a cart, they are not a necessary condition for abstract thought. Moreover, although abstraction can stifle and deaden, it may also have the capacity to disclose and enliven – the key point is to be ‘vigilant about [our] modes of abstraction’ (Stengers, 2018b, 111, paraphrasing Whitehead).

A starting point for inquiring into modes of abstraction is to note that its literal root is to ‘pull away’. For mathematicians at least, the allure of abstraction as beauty and refuge is significant. Hardy (1940: 45) writes: ‘When the world is mad, a mathematician may find in mathematics an incomparable anodyne’. So, in some instances, abstraction can be a source of great comfort. But it is not apolitical. Indeed, its use as ‘a secure platform for the rational examination of a world now rendered external’ – supporting, for example, Galileo's claim that the Book of Nature is comprehensible only in the language of mathematics – has led abstraction to being critiqued as ‘aligned with and reproducing disembodied habits of knowing, techniques of alienation, and failing to recognize corporeal difference’ (McCormack, 2012: 715–716).

This is just one reason to be wary of mathematical abstraction, many more could be mentioned. However, as McCormack (2012: 722; 726) emphasises:

the opposition between the lived and the abstract is insufficiently nuanced to grasp the ways in which abstraction participates in processes of world-making … the point of foregrounding the abstractions that frame space and time, particularly under capitalism, is not to replace them via an appeal to the immediacy of the concrete … [t]he task instead is to both reveal how abstraction works and to generate alternative abstractions as part of a necessarily critical praxis.

This is particularly relevant to the potential value of topology. As Section 3 aims to show, it can provide a language for encountering and engaging realities beyond what may be apparent or commonsense ¹⁰ – with the ‘contra’ of encounter, of meeting difference; and the ‘gage’ of engaging, pledging oneself, not doing away with dominant notions altogether. ¹¹

Abstraction and poststructuralism in topology

As Martin and Secor (2014: 435) note, the potential of topology ‘resides, we think, in how it is reworked through postructuralist theories of space’. What might this mean for abstraction in topology?

This is a matter, we suggest, of thinking beyond mathematical topology as something to be applied, which typically evinces structuralist abstraction by way of mapping an abstract mathematical framework onto a field of application (Paton et al., in press). Here, three features may be salient. First, abstraction is from without, employing an order inherited not from the situation at hand but from a pre-existing mathematical theory, its meaning and validity considered ab initio separate from context. Abstraction is therefore rigid, in the sense that it is disjunct from lived temporality. ¹² Finally, abstraction is totalising, implicating that which does not fit a model's lineaments as noise, error, or random variation; there is no room for the unabstracted.

But taking a cue from Long (2024), and thinking of mathematical topology instead as, say, a method for attunement rather than application, something different emerges around abstraction (see also Kanngieser and Todd, 2020). Rather than evicting flesh to arrive at skeletal structure, topology as employed by Deleuze and Lacan uses abstraction from within, eliciting structure as ‘immanent, which is to say that it is not separate from its effects’ and that it ‘operates within the scene’ (Martin and Secor, 2014: 432). ¹³ This mode of abstraction is more improvisational: no longer divorced from the energy of materiality, it is a means for finding a pulse or rhythm, a distillation of affect – a disclosure of spatial intension rather than extension (Figure 1). Finally, abstraction is partial, its value contingent on the specific situation at hand rather than derived from an appeal to the supposed verity of universal mathematical truth, so there remains room for other possibilities – just as illumination from the sun helps to find the moon in the night sky, but it is only when this is a sliver that the dark side of the moon is still visible.

Figure 1.

Abstraction as disclosure of affect, tension, life-force, in Kandinsky's drawings of dancer Gret Palucca (reproduced from Kandinsky, 1926; images in the public domain). Like a finger that points to the moon but is not the moon itself, ‘[c]ould it be that images do not stand for things but rather help you find them?’ (Ingold, 2011: 197).

As Whitehead notes, abstractions can ‘act as lures, luring attention toward something that matters’, their aim being ‘not to produce new definitions of what we consensually perceive and name, but to induce empirically felt variations in the way our experience matters’ (Stengers, 2008: 96, quotation marks removed; see also Segall, 2022). Whereas a structuralist abstraction of a dancer might seek to represent their spatial form by way of a geometrically faithful outline, the abstractions in Figure 1 could be called poststructural (in the sense of Martin and Secor, 2014) for their disclosure of affective intensity accommodating ‘an expanded empiricism, one in which the category of experience is no longer limited to the logics of spatial extension’ (Roberts, 2024: 9–11, emphasis added).

Topology between geography and mathematics

That which has no substance enters where there is no space.

– Tao Te Ching, verse 43 (Lao Tzu, 2015; translated by Stephen Mitchell)

We now turn to a number of key ideas in mathematical topology. As discussed in Section 2, when a concept is introduced below in a manner such as ‘a homeomorphism is a …’, it can be taken as a metaphorical statement: through what Sfard (1994: 47–49) calls the ‘metaphor of the ontological object’, the ‘is’ in the definition enacts the mathematical entity. There are always multiple ways of understanding the same term, thus mathematical entities necessarily exceed singular definitions (Thurston, 1994: 163–164). It is also worth noting, as Law (1999: 3) writes regarding actor-network theory, that a spokesperson presentation – ‘in mathematical topology, we do it like this’ – may give the impression of an immutable or unified body of knowledge, a fixed point. This is an illusion; we elucidate a handful of core topological concepts that are shared by many mathematics courses and textbooks in the subject, but our exposition is by no means the only way of configuring topology.

Preliminaries

Most topological ideas are presently framed in terms of ‘sets’ (introduced below). We therefore make use of this framing, but to a limited extent. Sets are metaphors, and – especially where they naturalise categories – not always helpful ones (Bowker and Star, 1999; Feyerabend, 1975; Wittgenstein, 1953). Taking the above cue from Stengers and Whitehead to be vigilant about our modes of abstraction, it is dubious to assume that the atomistic and objects-in-containers syntax of sets should be necessary for conceptualising space beyond mathematics, especially in a discipline with spatial vocabularies as rich as geography. Although many mathematicians rely on sets to organise thought and structure arguments, the ways in which mathematicians feel, visualise, communicate, and work with topological concepts goes far beyond their set-theoretic formalism. In particular – and unlike several other contemporary articulations of topology in the social sciences (e.g., Phillips, 2013; Sha, 2012) – we avoid using symbolic notation from set theory, since this often serves to gatekeep mathematical knowledge.

Sets and infinities

A set is a collection of things, usually called ‘elements’ or ‘points’. In general, sets are considered to be without explicit spatial form (Munkres, 2000 [1975], 3). However, they are often visualised schematically as in Figure 2(a) – sometimes as discrete points, other times as a ‘block … in which the discernibility of points disappears’ (Deleuze and Guattari, 1987: 294).

Figure 2.

(a) Two sets, one drawn as four discrete points enclosed in a container, the other as a continuum of points (indicated by shaded lines). (b) Venn diagram depicting the union and intersection of two sets. (c) Circle and line intersecting. (d) Blob and loop contained in a torus.

One can produce new sets from given sets – or, depending on how agency is considered, sets can produce new sets. The union of two sets is the collection of elements in either one or the other or both, illustrated by the area shaded by horizontal lines in Figure 2(b). The intersection of two sets is the collection of elements common to both, shown as the area shaded by vertical lines in Figure 2(b). For example, the union of the circle and line in Figure 2(c) is the whole of this diagram, and their intersection is the two points indicated. If two sets partially overlap, as in Figure 2(b) and (c), then their union could be said to be more than one but less than two, and their intersection more than zero but less than one.

One set is a subset of another set if all elements of the first set are also elements of the second set: for example, in Figure 2(d), the blob and loop are both subsets of the torus. Thus, any set is a subset of itself, and the intersection of two sets is a subset of each. The ‘empty set’, the set with no elements, is a subset of every set. ¹⁴

A set is an unmanifest spatiality. Topology actualises a set's spatial potential (see below), but in the context of set theory, two sets are considered equivalent simply if they have the same size – the same amount or intensity of unmanifest spatial potential – which here means that a one-to-one correspondence can be drawn between their elements. Sets can have finite or infinite size. An infinite set is countable if it has the same size as the so-called ‘natural numbers’ (0, 1, 2, 3, and so on), otherwise it is bigger and called uncountable. The smallest of the uncountable infinities is the continuum, also called the ‘real numbers’ (see Dauben, 1983 for more details). ¹⁵

The continuum is often visualised as the set of points which fill the number line, though its potentiality can also fill the Euclidean plane, and other spaces which seem much larger (Supplementary Figure 1; see also Earl, 2019: 19). Mathematicians occasionally work with bigger, more intense infinities. Nonetheless, sets whose sizes are finite, countably infinite, and the continuum, usually provide enough variation in spatial potential for much of mathematical topology.

Topological spaces and open sets

In the spirit of attunement described in Section 2, a heuristic for abstraction in mathematical topology is to distil certain qualities which are immanent in a more tangible setting, and then take these qualities as defining features of a new, more abstract setting (Armstrong, 1983: 12; Munkres, 2000: 75). Thus while topology moves beyond the metrics and geometries of Euclidean spaces, the latter provides a prototype for a topological space, a place for ‘tuning’ the abstractions of topology (cf. Stengers, 2008: 106).

In mathematics, a topological space (or just space, in the context of topology) consists of two things. First, a given set, which could be called the potential space, since its spatiality is considered unmanifest. ¹⁶ Second, a given collection of subsets of the potential space, such that:

The union of any number of subsets in the collection is also in the collection.

The intersection of any two subsets in the collection is also in the collection.

The potential space itself, and the empty set, are both in the collection.

These subsets of the potential space are called open sets, and the given collection of open sets is called a topology for the potential space. A potential space, together with a topology, make a topological space. Two elements of a space are usually considered close, inseparable, or indistinguishable, if every open set which contains one element also contains the other. ¹⁷

Now, it is not obvious at all (including for mathematicians ¹⁸ ) why this formulation has anything to do with space in a familiar sense, or spatial concepts such as connectedness and continuity. But this can be seen in its favour – its strangeness rebuffs easy assimilation into Euclideanism. As we attempt to show through the rest of Section 3, the above offers a minimalist language which augments rather than replaces mathematical vocabularies of space. On the one hand, it can be used to elicit contingencies and nuances of Euclidean spaces or other metric spaces, and thereby challenge their essentialism. On the other hand, its lack of intuitive substance affords it the possibility of enacting space in areas of mathematics where there is no apparent spatiality (Aleksandrov et al., 1999, Vol. 3, 135–136).

Of relevance here is Merriman (2012), who questions whether time, space, time-space, and the like – rather than movement, rhythm, sensation, or energy – should be ‘primordial, foundational, ontological’ principles through which geographers organise their accounts of human and nonhuman life. We share this doubt, and note that topology's language of open sets is a way of patterning difference which makes no explicit recourse to space in an everyday sense. ¹⁹ Although many of the arguments in this article are framed in terms of ‘space’ and ‘spatiality’, we treat these terms with a degree of descriptive ambiguity. For much of Section 3, ‘time’ or ‘temporality’ could also work, and we avoid terms like ‘space-time’ because (in consonance with Merriman, 2012) we are not aiming for a totalising account of difference. ²⁰

Euclidean space

We now aim to bring to life the spatial work that open sets do by describing and illustrating: (i) how they constitute Euclidean space; (ii) three ways in which topology takes a relational approach to space; and (iii) how topological and geometric approaches to Euclidean space can differ. This is then drawn into dialogue with Hinchliffe et al. (2013; 2017). In what follows, Euclidean lines, planes, volumes, and the like, are inclusively referred to as Euclidean spaces; and a ‘globe’ refers to a spherical volume, whereas a ‘sphere’ is taken to be hollow.

Euclidean space with the standard topology

What geometers call ‘Euclidean space’, topologists call ‘Euclidean space with the standard topology’. Here, the simplest open sets are formed by considering all points within, but not equal to, a particular distance from a given point (Figure 3(a)). Other open sets in the standard topology are less symmetric (Figure 3(b)). That these are open sets and that they constitute familiar Euclidean space is not obvious – it takes some work to show these do indeed fit the formulation of a topological space (Section 3.1.2, ‘Topological spaces and open sets’) – but we bracket this here to move on with the argument.

Figure 3.

Open sets in Euclidean space with the standard topology. (a) In a Euclidean line, this is an interval without the boundary points; in a Euclidean plane, this is a disc without the boundary circle; in a Euclidean volume, this is a globe without its outer shell (its boundary sphere). (b) A more general open set, consisting of three open pieces. (c) The sequence 1/2, 2/3, 3/4, and so on, stays within the open set, getting infinitely close to 1, though 1 itself is outside the set.

All open sets in Euclidean spaces share an important quality that helps intuit how open sets work in less familiar spaces. It is that any point in an open set can move, vibrate, wiggle a little bit, and still remain within that set. Hence, open sets do not have boundary points. In the language of Hinchliffe et al. (2013), open sets intersect in borderlands rather than borderlines. A strange consequence is that a sequence of points in an open set can get infinitely close to a given point even as this ‘limit point’ may itself lie outside the open set (Figure 3(c); see Earl, 2019: 80–82).

The pivotal idea here is one of conditionality: the illustrations in Figure 3 are Euclidean spaces with the standard topology. A potential space can be actualised with different topologies. Assuming conventional set-theoretic axioms, two topologies which are possible for any set are: (1) the ‘indiscrete topology’, in which there are no open sets other than the empty set and the whole potential space, so there are no open sets which can distinguish points; (2) the ‘discrete topology’, in which every subset is an open set, including the points themselves. These are two polar opposites of how any potential space can manifest spatially: the former is like an undifferentiated lump, a homogeneous unity, considered the ‘coarsest’ topology; the latter is like a cloud of dust, a total discretisation, considered the ‘finest topology’ (Earl, 2019: 79; Munkres, 2000: 77). Euclidean space with the standard topology is somewhere in between.

Relationality in Euclidean space

The first relational point, therefore, is that the relations between the open sets perform the space. Neither the potential space, nor an open set on its own, disclose spatiality. Rather, it is the way open sets intersect and form unions with other open sets that manifests the space. Indeed, with one topology for a potential space – that is, with a specific collection of subsets satisfying the three conditions for a topological space (Section 3.1.2) – two elements may be distant. Yet with another topology, the same two elements may be so close they are indistinguishable, as in the indiscrete topology mentioned above. So, in the words of Law (2002), spatialities are enacted, not a given.

Second, open sets are not ‘open’ in themselves, but only in relation to the potential space and the other open sets. Following the diagram in Jänich (1984: 10), in Figure 4(a) an open set of a line is no longer open when the line is considered embedded in a plane, because any point on the line which vibrates a little up or down leaves the set.

Figure 4.

(a) An interval without endpoints is open as a subset of a line, but not as a subset of a plane. (b) Squiggly line and plane.

Third, topology makes explicit what geometry leaves implicit, which is that subjectivity is involved in manifesting space – or, more radically, subjectivity and space manifest each other. The line and plane in Figure 4(a) are topologically equivalent ²¹ to the undulating line and plane in Figure 4(b). Just as Kandinsky's gaze is present in the diagrams of Palucca's dance (Figure 1; see also Mulvey, 1975), so too do the spaces diagrammed in Figure 4 therefore trace the knower's mind in the form of the known, just one visual expression of (often infinitely) many possible others. What is visibly present implies a much greater otherness.

In more detail: consider how in Figure 3(a), the Euclidean line looks full, in the sense that it is filled with points, whereas the Euclidean plane and volume seem empty – placeless voids framed by coordinate axes. But this is only because diagramming on paper affords the ability to draw a dark line and indicate open sets on that line by making use of the light-coloured paper outside the line, whereas the diagrams in Figure 3(a) leave the fullness of Euclidean two-space and three-space implicit, in order to make visible one of its infinitely many open sets.

Hence there is no ‘God trick’, in the words of Haraway (1988). No total view from outside. Space discloses itself only partially (Abram, 2010), because what one discerns is not separate from one's concerns. ²² A specific terrain may reveal itself very differently from one human person to another, or to the sensorium of an oak drinking sunlight with its leaves even as its roots and rootlets are searching out and sipping moisture within the soil of the apparently ‘same’ landscape. It is only in classical geometry that Euclidean space is a terra nullius for inscription by human mathematicians, since topologically it can be far more intense than just a void or set of points. Akin to how Deleuze and Guattari (1987: 382) evoke a desert: ‘visibility is limited … there is an extraordinarily fine topology that relies not on points or objects but rather … on sets of relations…. It is a tactile space, or rather ‘haptic’, a sonorous much more than a visual space’. ²³

Diagramming disease situations

Joining up with threads from the Introduction, we note that Hinchliffe et al. (2017: 52–81) emphasise the importance of finding topological diagrams which may engender an understanding of how: (i) disease situations exceed individual sites, but are not transcendent to them; (ii) these situations are a tension between multiple spatialities, including extensive and intensive narratives of disease; (iii) pathogens do not move across space, rather pathogenicity is itself a spatial configuration. There are layers of resonance to note here between these concerns of Hinchliffe et al. (2013, 2017) and the language and imagery of open sets.

Specifically, given that pathogens, considered topologically, are spatial configurations themselves – ‘a set of potential forms actualised only in relation to the environments engaged’ (Hinchliffe et al., 2013: 538) – there is scope for diagramming them as open sets, rather than points, of Euclidean (or similar) spaces with the standard topology. While these open sets sit inside an extensive geometric space, they also enact it relationally as an intensive topological space. Moreover, as with pathogens, a particular open set exceeds its constituent points (through its relations with other open sets) but is not transcendent to them.

Closed sets

In anticipation of the rest of Section 3, there is one additional concept to mention here. A closed set is the negative space around (called the ‘complement’ of) an open set, a kind of figure-ground reversal (Munkres, 2000: 93). Closed sets can also be enacted as the union of a set and its boundary points (called the ‘closure’ of the set; Munkres, 2000: 97). For example, the closures of the open sets illustrated in Figure 3(a) are: a line with its two endpoints included; a disc with its enclosing circle; a globular volume with its spherical shell. Closures and complements are operations that typically enact different closed sets: whereas the closure of the open disc in Figure 3(a) would be the disc together with its boundary circle, the complement of this open disc would be the entire Euclidean plane minus the open disc, the boundary of which is the same circle that encloses the open disc. So, closed sets, which do have borderlines, are in a sense dual to open sets; they are co-constitutive. ²⁴

Folding space

One way in which topology challenges Euro-American spatial notions is by demonstrating Euclidean space as relationally enacted, as discussed above. However, topology's language of space is such a radical departure from modernist ways of thinking that, in its minimalist formulation (Section 3.1.2), it can be difficult to grasp and operationalise. Mathematicians have thus developed methods for constructing and diagramming spaces that employ Euclidean spaces (with the standard topology) precisely in order to think beyond them. We describe here one such technique, called ‘quotienting’, which we then draw into dialogue with topological ideas in Agamben (1998) and in how his approach has been taken up in human geography.

Quotient spaces

Folding, also termed gluing, or suturing, is central to mathematical topology, and arises in the practice of making quotient spaces (Jänich 1984: 43–49). The term ‘quotient’ refers to the arithmetic operation of ‘division’, which could equally mean ‘bringing together’: consider, six can be expressed as two times three, and dividing six by two is to produce the quotient, three, by making the two come together as one (Supplementary Figure 2). ³⁴ Just so, a quotient space in topology is formed by first stipulating identifications between different points, and then considering identified points as the same point, thereby cancelling, or ‘quotienting’, their difference. ²⁵

Recall the aforementioned closures of open sets in Figure 3(a): a line including its two boundary points; a disc including its boundary circle; a globe including its boundary sphere. As illustrated in Figure 5, bending a line and then gluing its two boundary points enacts a circle; stretching a disc and suturing together its boundary circle enacts a sphere; what about gluing together the entire boundary sphere of a globe, but leaving the globe's inside topologically unchanged? ²⁶ Following the recursive pattern here, in which the quotient space of one set enacts the boundary of the next, one might say that the quotient space of the globe (called a ‘three-sphere’) is the boundary of another, altogether unfamiliar, set (known as a ‘four-ball’).

Figure 5.

(a) Quotienting a line by its two boundary points (signified in red) performs a circle, also known as a one-sphere. (b) Quotienting a disc by its boundary circle (signified in red) constructs a sphere, also known as a two-sphere. (c) Quotienting a globular volume by its boundary sphere makes a three-sphere. Lived experience handling materials such as clay may help conceptualise (a) and (b) – here, it is as though topology is like geometry prior to being fired in a kiln (cf. Nemirovsky et al., 2023).

The three-sphere and four-ball are beyond the purview of Euclideanism. How to think about them? One way is to use a temporal narrative inspired by the above recursive sequence, and to observe the traces they leave as they move through Euclidean volumetric space (Figure 6). Another way to understand a three-sphere, given the bodily basis of mathematical reasoning, is to stay with the spatial metaphor of a globe and instead move from a visual to a haptic sense. Here, visibility is limited, and imagination is situated within (rather than looking at) the object, feeling one's way around the globular volume with the knowledge that its boundary sphere has somehow been folded into a single point without topologically changing the globe's interiority. ²⁷

Figure 6.

Traces of objects passing through Euclidean spaces: (a) When a disc (whose boundary is a circle) moves through a Euclidean line, it appears as a line segment that expands, contracts, and vanishes. (b) When a globe (whose boundary is a sphere) moves through a Euclidean plane, it appears as a disc that expands, contracts, and vanishes. (c) Thus, one may expect that when the next step in the recursive sequence (a four-ball, whose boundary is a three-sphere) moves through a Euclidean volume, it appears as a globe that expands, contracts, and vanishes.

It is challenging to communicate these spatialities without the improvisational gestures and dynamic diagramming that animates much of mathematical topology in practice. In any case, the limits to understanding space in terms of spatial extension, that took effort to elicit in Section 3.2 (‘Euclidean space’), now become conspicuous.

Diagramming a state of exception

That abstraction is an instrument for colonial violence, entailing erasures, indifference, and dehumanisation, is well known – the role of acronyms used in military mapping by the United Kingdom's Ministry of Defence is a case in point (Ingold in Olwig, 2019, xv). In a discussion which inspires our inquiry and echoes Section 2, Doolittle (2018) describes how non-Euclidean geometry and topology can also be means for emancipation, working to bring to light and undo spacings of human control and oppression. It is in this second manner that Agamben (1998) invokes topology to examine how a ‘state of exception’ is created.

A state of exception is a spatial ordering which includes within its rule precisely what it places outside the realm of law, so that those living under it are incorporated as objects of sovereign power even as they are excluded from being its subjects. Agamben (Agamben 1998: 28) employs a Möbius strip to articulate this indistinction or superposition of outside and inside, or what he calls physis and nomos. Gregory (2004) uses this topological approach to evidence how the Israeli settler-colonial state – supported by British and American interests in particular – has splintered and deformed the landscapes of Palestine, rendering its villages as spaces of exception in which ‘human life is included in the juridical order … solely in the form of its exclusion (that is, of its capacity to be killed)’ (Agamben, 1998: 12). In her seminal work on the afterlives of Transatlantic chattel slavery, Sharpe (2016: 16) writes similarly: ‘to live in the no-space that the law is not bound to respect, to live in no citizenship.’

Giaccari and Minca (2011), in their study of Nazi spatial practices, draw on Agamben (1998: 19) to detail how, in granting

the unlocalizable a permanent and visible localization, the result was the concentration camp … the camp is topologically different from a simple space of confinement.

Thus, while the Möbius strip provides a way of thinking about the unlocalisable, it leaves untouched the question of how this zone of indistinction between inside and outside is localised among more commonplace spatialities, as a threshold where ‘the topographical and the topological came together’ (Giaccari and Minca 2011: 7).

Consider, then, how a Möbius strip might be localised. In Figure 7, an open disc is cut from a sphere, a process which mathematicians may call ‘excision’ or ‘surgery’ (note the affective language, and its consonance with Gregory's descriptions of torsion, severation, and fracture). The boundary of the resulting hole in the sphere is a circle which, topologically speaking, is also the shape of the Möbius strip's boundary (Figure 7). So, suturing the Möbius strip to the sphere along this circle produces a space which, while resisting visualisation from the perspective of a detached observer, nonetheless localises the superposition of inside and outside in a smooth transition to the surrounding surface of the sphere. If this space is paradoxical, it is because a Euclidean imagination cannot enact this process without tearing the Möbius strip or seeing it intersect the sphere away from the boundary circle. But from the situated perspective of the sphere's surface, there is no break or oblique intersection, only an indeterminate zone in which inside and outside become seamless – indeterminate because the boundary can stretch and deform without changing topologically, so if one is within or without it, near or far from it, remains unknown. ²⁸

Figure 7.

(Top row) Stretching a Möbius strip into a shape resembling a snail shell, in order to show how its boundary is a circle. The visible half of this boundary circle is signified in black in the rightmost shape; rotating this Möbius shell around a vertical axis reveals the other half of the circle. The red arrows indicate the shapes’ interiorities: a solid arrow is on a visible surface facing the reader; a dashed arrow is on a hidden surface facing the reader; a dotted arrow is on a surface facing away from the reader. (Bottom row) Excising a disc from the surface of a sphere, then suturing the Möbius shell along its boundary circle to the hole's circular edge.

We wish to tread with due solemnity here. Whether this spatial metaphor should hold up to scrutiny, regarding its value for understanding spaces of exception, is not for the present authors to judge. Although we are not alone in mobilising topology to address spatial practices of colonial omnicide (see also Arora et al., 2025), we emphasise the remark of Gregory (2004: 248) that the ‘world does not exist in order to provide illustrations of our theories’.

Homeomorphism

This final part of Section 3 aims to nuance and critically engage how topology is employed to understand invariance – the ways in which things endure through change. It is here that one typically encounters ideas of ‘homeomorphism’, a kind of spatial transformation which, roughly speaking, alters geometric appearances but leaves topological relations unchanged. Below, we look at how homeomorphisms are deployed in the social sciences and in mathematics, in order to: (i) evidence how discussions of spatial change and continuity may retain implicit assumptions of a background Euclidean space and time; and (ii) suggest ways of patterning invariance in contexts attuned to divergence or incommensurability.

To begin with, stretching, twisting, and bending, without tearing or breaking, are how homeomorphic actions are often conceptualised in geography (e.g., Secor, 2013; Shields, 2012). For sure, these transformations move beyond Euclidean geometry's reliance on fixed metric distance as a necessary condition for spatial invariance. But they also retain the primacy of Euclideanism, because these actions implicate an encompassing space in which the deformations occur – and thus the possibility of a detached and unchanging observer who can view a transformation happening from outside the space being transformed.

Law (2002: 95), noting that ‘questions of spatiality and object continuity are settled together’, uses a diagram to address this (Supplementary Figure 3, reproduced from Law, 2002: 94). His point is that the apparent inability to deform the first pair of circles into the second, without cutting open the larger circle, rests on an assumption that the page is the environing space in which this deformation must happen. If instead one rotates the smaller circle up and out of the page, then back down again, the second diagram can be produced from the first without breaking the larger circle. Hence, whether these two pairs of circles are deemed equivalent as objects of knowledge depends on whether a subject of knowledge see the circles as having to remain within the page or not. In short: to perceive continuity is to participate in continuity.

The conditions of possibility for invariance have expanded from the page to include the medium of air through which the hand moves. Yet the idea remains of an a priori ambient space in which such a transformation must happen. One can go further.

In mathematical topology, two spaces are equivalent, or homeomorphic, if there is a one-to-one correspondence between their points which induces a one-to-one correspondence between their open sets (Armstrong, 1983: 34; Munkres, 2000: 105). In less atomistic terms, this effectively means that the open sets constituting one space may appear different in form to the open sets constituting the other, but the relations (of intersections and unions) through which they collectively enact the space is the same.

This formulation is highly abstract and not easily operationalised. Indeed, it takes some work to show that ‘deformation without rupture’ (Law, 2002: 95), and transformations in which there are ‘no great breaks or disruptions’ (Law and Mol, 2001: 614), like those mentioned by geographers above, can be examples of homeomorphisms in mathematical topology. Nonetheless, this approach also affords ways of patterning sameness across differences which are not reconciled through stretching, bending, twisting, and the like.

How to make sense of this – quite literally, how to connect this to the senses? Noting the discussion of abstraction in Section 2, a diagram can help lure one's attention to what matters. Following Armstrong (1983: 11), see Figure 8 for two ways of drawing a Möbius strip. They are not continuously deformable into each other, but they are homeomorphic. This can be illustrated in two steps. First, topologically speaking, twisting once and twisting thrice are equivalent actions, so the shapes in the top left and top right of Figure 8 are the same. Second, although the suturing action that joins together the two arrowed edges for the shape on the left is not a homeomorphism (to reverse the action would be to break apart the Möbius strip), it has the same effect on the open sets of the once-twisted square as does the suturing action for the thrice-twisted square. Hence, the changes undergone by the open sets in the left trajectory are topologically equivalent to those undergone in the right trajectory, so the resulting two Möbius strips are homeomorphic.

Figure 8.

Two Möbius strips, one with a single twist, the other with three twists (also called half-twists). The arrows are superfluous to the spaces and indicate the alignments by which the edges are to be quotiented (in other words, joined). In this figure, the square can be thought of as a piece cut from a Euclidean plane. This means that its open sets are derived from the plane: a set is open in the square if it is the intersection of the square with an open set in the plane. Hence the square and the plane have similar blob-like open sets, with one key difference: whereas open sets of the plane have indeterminate boundaries, open sets of the square may include the square's edges, since from the point of view of the square, there is no space beyond its edges for points on an edge to vibrate or wiggle into (cf. Figure 4(a)).

This homeomorphism illustrated between the Möbius strip with one twist and three twists is a relation of equivalence that is constituted by divergence. Indeed, it arises from an affinity not through continuous deformation in the present, but from a divergent pair of transformations the square undertook in the past. Drawing on Gregory (2004: 256), one might term this a narrative of connective dissonance, in which ‘connections are elaborated in some registers’ (tracing the trajectories of Figure 8 in parallel) ‘even as they are disavowed in others’ (no homeomorphic transformation through continuous deformation within an ambient Euclidean space).

There are layers of resonance here with the approach to physics elaborated by Bohr and Barad. Akin to the ontological indeterminacy of physical objects and spaces prior to measurement (Barad, 2007: 117–118), ²⁹ topological spaces also have an indeterminacy prior to being diagrammed and made visible. Drawing a Möbius strip with a single twist, as is the usual depiction, elicits this one configuration among infinitely many homeomorphic others – with three twists, five twists, and so on, all of which may be considered to be in superposition with each other. Space, therefore, has many faces.

But then so too does time: the homeomorphism illustrated in Figure 8, and the temporal language of past and present employed above, shows up the linearly unfolding temporality often implicit in conceptualising spatial transformation. Our point is emphatically not that questions of spatial change and continuity in the social sciences should be approached with allegiance to a mathematical formulation of homeomorphism. Rather, there is more to be explored by critically examining the multiple conditions of possibility for invariance, and accommodating situations in which ‘[t]here is no overarching sense of temporality, of continuity, in place … no smooth temporal (or spatial) topology connecting beginning and end’ (Barad, 2010: 244).

Conclusion

Morgan creates a non-creator subject, a dissolved self. Not so he will fade away, but in order to get clean water flowing everywhere.

– Marianne de Laet and Annemarie Mol (2000: 251)

The implications of the previous section are uncanny. In reviewing mathematical topology through spatial vocabularies of (more-than-)human geography, as we have attempted to do, mathematics emerges as an unlikely ally to a number of important insights on space in the social sciences. It can help evidence, for example, how spaces are enacted, relational, patchy, and contingent; have more intensity and interiority than supposed by Cartesian logics of spatial extension; and only disclose themselves partially, so there is no totalising ‘view from outside’. Why, then, is this presentation of mathematics so unfamiliar? Could it be that mathematics itself has these same above characteristics which topology elicits of space – so that one does not necessarily need an altogether ‘new’ mathematics to approach uncertain, messy, wild, ambiguous, unruly phenomena – but that its conventional usage somehow hides its many, sometimes discordant, faces? And what might this mean for topology in geography?

Beyond application

We contend that mathematical topology's rich and heterogeneous potential for thinking space dissipates when it is considered something to be ‘applied’ to ‘the real world’. Recalling Section 2, this implies an ab initio separation – and, therefore, creates the conditions for a one-way, hierarchical relationship – between mathematics as a language of description, and what it is used to describe. As one mathematician notes: ‘Much of applied mathematical analysis can be summarized by the observation that we continually attempt to reduce our problems to ones that we already know how to solve’ (Keener, 2018, xi). Although an effective heuristic for knowledge production within academic mathematics, the result is that application, in its prevailing logics, moulds the world in the image of mathematical knowledge as a precondition for studying the world with mathematical knowledge. Indeed, de la Cadena and Blaser (2018: 3) note how allegiance to a singular, privileged way of knowing ‘cancels possibilities for what lies beyond its limits’. ³³

This reversal of priorities – conforming context to abstraction, rather than taking abstraction as immanent and attuned to context – that engenders such a tautology is the trap of application, which can lead mathematics to play (as Pink Floyd might say) a lead role in a cage. ³⁰ Mindful of this, in Section 2 we reviewed the bodily and metaphorical basis of mathematical reasoning to suggest that a shift from application to attunement can offer a more resonant, corporeal, and improvisational use of abstraction, one which is committed to concepts emerging as partial perspectives from within a context rather than being imposed as totalising frameworks from without.

This laid the ground for our approach in Section 3. Eschewing the symbolic notation commonplace in mathematical topology textbooks, we sought to describe a number of key concepts – including sets, topological spaces, openness and closedness, quotient spaces, and homeomorphism – in a manner more attuned to the spatial lexicons of geographers. A first implication was that Euclidean space, topologically considered, is already constituted relationally (by open sets) and contingent on human subjectivity. This could offer tools for diagramming spatialities which require a consideration of both topographical and topological effects, such as those discussed in Hinchliffe et al. (2013; 2017). The tension between topography and topology was again mobilised in the notion of a quotient space, where fragments of Euclidean spaces are folded, glued, or sutured to produce unfamiliar spaces which challenge logics of spatial extension and the view from nowhere. This suggested a way to describe how a ‘state of exception’ (Agamben, 1998; Gregory, 2004), in one sense paradoxical, could still be situated in topographical space. Finally, the framing of homeomorphic transformations in terms of open sets helped expand the conditions of possibility for understanding invariance (how things may endure through change), specifically by finding a way of patterning difference which is less reliant on background assumptions of a transcendent space and time.

Object and subject

For mathematical topology to come into receptive dialogue with other realities and ways of knowing, rather than replicating its own reality and performing a one-world world in which other worlds do not fit (Law, 2015), we found in writing Section 3 that there was an important point to address regarding the entanglements of knowledge and social order (Bloor, 1991; Jasanoff, 2004).

Namely, despite the conceptual richness and subtleties of mathematical topology, the unitary social order which often frames Euro-American mathematical knowledge enacts a regional topology (in the language of Mol and Law, 1994). According to this logic, there is an inside, the modern subject, and an outside, the mathematical object (Rosen, 2006). The assumption is that neither is constitutive of the other, so that mathematics simply ‘reflects the laws of the material world around us and serves as a powerful instrument for our knowledge and mastery of nature’ (Aleksandrov et al., 1999, iii).

Taking mathematics as a detached representation of reality, rather than an intervention or participation in realities, would truncate topology's relevance and scope in geography. Consider Law's observation that not all flows belong to fluid space – think of early uses of topology in geography to model flows of migration and transportation (Merriman, 2022: 77) – because fluid topology is not separate from fluid subjectivity (Law, 2002: 100–101). Fluid objects seem to resist being known by a unitary social order, or a controlling, patriarchal subjectivity. Indeed, in their study with the bush pump, de Laet and Mol (2000) evidence the connection between the success of this technology, and the refusal of Morgan (the designer) to take credit for its invention or to patent it, and hence to create a fixed point of centralised accumulation. Consonantly, as Hinchliffe et al. (2013: 539) write regarding biosecurity:

If disease can be diagrammed topologically, so too can the knowledge required to understand and act on disease … understanding a disease threat relies on loosely coupled knowledge practices, rather than a tightly orchestrated, centralised surveillance system.

This is not to say that fluid space is somehow ‘right’ and Euclidean or network space ‘wrong’ – thinking improvisationally with Euclidean space may be more effective for understanding alterity and nonconformity than thinking rigidly in the name of fluid space – but rather that a ‘sensibility for complexity is only possible to the extent that we can avoid naturalizing a single spatial form, a single topology’ (Law, 1999: 7). Inspired thus, we have attempted to describe topological concepts in accordance with standard mathematics textbooks, but with a subjectivity and language more closely aligned with geography. The result – which became apparent for us only when writing this article – has been that a less unitary, more socially attuned subjectivity necessitated writing objects with greater plurality and ambiguity.

This touches a rather foundational question: what kind of knowledge does one value? Mathematics is often employed as a definitive and unconditional means to classify and divide, to adjudicate with transcendent authority over what ‘is’ and ‘is not’. ³¹ But if topology is to move beyond a curated academic setting and bring rigour and precision to messy spatial contexts, this approach to knowledge production – and what rigour, precision, and messiness really mean – may require rethinking. In particular, under the rubrics of application articulated above, the ambiguities and pluralities in Section 3 may be seen as a technical shortcoming, or a failure to define terms clearly. And this may entirely miss another possibility: that we have been rigorous and precise, but are working with phenomena which require a degree of descriptive indeterminacy and multiplicity in order to become visible – worlds which cannot be known if they are rendered singular and categorised, like a specimen pinned to a board and labelled. ³² Indeed, it is in the milieu of the social sciences, not mathematics, that a fuller understanding of the effects of multiple and intersecting topologies has been made possible.

Hence, topology rebuffs modernity's object/subject separation, and does so in multiple ways. Recalling the quotation which began this article, there is a further practical implication to be mentioned here. Rather than just using topological concepts as mute objects of knowledge to be deployed according to the dictates of human volition, something may also be gained by asking if topology has hidden faces to reveal to its subjects of knowledge. Strange though it may seem, to take topology seriously is to find the figure of the knower in what is known; to see one's subjectivity emanating back from the diagrams whose ink is still drying on the page; to consider that, as the human does topology, so too does topology do the human.

Supplemental Material

sj-pdf-1-dhg-10.1177_20438206251406736 - Supplemental material for Topology beyond application: Drawing social and mathematical worlds into rhythm

Supplemental material, sj-pdf-1-dhg-10.1177_20438206251406736 for Topology beyond application: Drawing social and mathematical worlds into rhythm by Siddharth Unnithan Kumar, Andy Stirling, David Abram and Sanjiv Kumar in Dialogues in Human Geography