Given this definition of two homeomorphic spaces,(adsbygoogle = window.adsbygoogle || []).push({});

Deﬁnition 1.7.2.Two topological spaces X and Y are said to be homeomorphic if there are

continuous map f : X → Y and g : Y → X such that

f ° g = I_{Y}and g ° f = I_{X}.

Suppose I know f and g are both continuous. Would it be safe to assume then, that if

f ° (g ° f) = f, X and Y are homeomorphic?

Here's my reasoning:

f ° (g ° f) = f implies g ° f = I_{X }and due to the associativity of a composition,

f ° (g ° f) = (f ° g) ° f = f or f ° g = I_{Y}.

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# Definition of two homeomorphic spaces

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