Four properties of M-fuzzifying interval spaces, namely, M-fuzzifying geometric property, M-fuzzifying modular property, M-fuzzifying Pasch property and M-fuzzifying sand-glass property, are introduced and characterized. Those properties together with M-fuzzifying Peano property and M-fuzzifying JHC property are closely related to each other. For example, an M-fuzzifying idempotent Peano-Pasch interval operator possesses M-fuzzifying geometric property and M-fuzzifying sand-glass property. In addition, in an M-fuzzifying modular space, M-fuzzifying Peano property, M-fuzzifying Pasch property, M-fuzzifying geometric property and M-fuzzifying sand-glass property combined with JHC property are equivalent.
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