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A, B are metric spaces.

Question: If A, B are homeomorphic then A is topologically homogeneous iif B is.

I know that if A and B are homeomorphic then there exist a function $g: A \to B$ continuous, biyective and which $g^{-1}$ is also continuous (an homeomorphism). And topologically homogeneous means that I can take any $(a,b) \in A$ where $\exists$ an homeomorphism $f: \ A \to A$ such that $f(a)=b$.

I'm confused about how to think about it.

If B is not topologically homogeneous it means that there exist some $(a,b)$ such that $f(a) \ne b$ and B is homeomorphic to A, assuming that A is topologically homogeneous means that $\exists \space f(a)=b$ for any $(a,b) \dots$ I'm trying to make a contradiction, could be?

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    $\begingroup$ I assume $f$ is supposed to be a homeomorphism in the definition of homogeneous? Please update your question if this is true. $\endgroup$
    – JonathanZ
    Commented May 2, 2022 at 4:42
  • $\begingroup$ Oh, and as to your main question: I wouldn't use contradiction. Do you know how to use a homeomorphism to turn a function on one space into a function on the second space? $\endgroup$
    – JonathanZ
    Commented May 2, 2022 at 4:45
  • $\begingroup$ @JonathanZsupportsMonicaC yes, f is supposed to be a homeomorphism. I update it. What do you mean with "how to use a homeomorphism to turn a function on one space into a function on the second space?" $\endgroup$ Commented May 2, 2022 at 5:01
  • $\begingroup$ If you have a (continuous) function defined on $A$ it uniquely determines a (continuous) function on $B$, and conversely. If you haven't seen this before, maybe you can figure out how to do so by looking at the domains and ranges of the various functions you have. Remember, the composition of two continuous functions is a continuous function. $\endgroup$
    – JonathanZ
    Commented May 2, 2022 at 7:03

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