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Given two topological spaces with X={d,e,f} and Y = {d, 2,3,8}. Explain why they are not...

Given two topological spaces with X={d,e,f} and Y = {d, 2,3,8}. Explain why they are not homeomorphic.

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Answer #1

A function f from topological space X to other topological space Y is called homeomorphisms if f is continuous, one one, onto with continuous inverse.

Here X={d,e,f} and Y={d,2,3,8}.

X has 3 elements and Y has 4 elements.

Therefore, there does not exist any map between X and Y which is both one one and onto.

Because for a function to be one one and onto, both the spaces should have same cardinality.

Therefore, topological spaces with X and Y are not homeomorphic.

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