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a few days ago a question arose.

We know that 2 spaces are homeomorphic if there is a continuous and bijective function with a continuous inverse, there are also other theorems that assure us that something is homeomorphic, but let's start from that initial definition.

Intuitively this is that we can deform one space into another in a continuous and reversible way. Is there a way to count the number of homeomorphisms that exist between 2 spaces?

I've seen some posts where they count something like the number of homeomorphisms between $ G_\delta $ and Cantor's set, as you can see here. In a basic case for example, we have that for the circle the number of homeomorphisms is at least $ \mathfrak{c} $ (the rotations), at first I thought that this was related to the degree of homogeneity of the space, but I am realizing that no, I would like to know if someone knows what this area is called and if it is possible that there is a space that is homeomorphic only with itself.

First of all, Thanks.

Haus
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    Maybe this can help: Counting the homeomorphisms between homeomorphic spaces is the same as counting automorfisms of the first space. – Asinomás May 18 '21 at 14:55
  • Thank you, you are right about that. All comments are well received. – Haus May 18 '21 at 15:03
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    The number of such homeomorphisms is usually huge, for standard metric spaces. But, we might be able to classify them into a finite number of classes up to homotopy. For example, the homeomorphisms of $S^n\to S^n$ are, up to homotopy, $\pm 1,$ depending on whether the homeomorphism is orientation-preserving or not. – Thomas Andrews May 18 '21 at 15:04

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Maybe this can help. Given two homeomorphic spaces $A$ and $B$ the homeomorphisms from $A$ to $B$ are in direct correspondance with the automorphisms of $A$. This is because we can take a fixed automorphism $g$ from $B$ to $A$ and then the map that sends a homeomorphism $f:A\rightarrow B$ to $g\circ f$ is a bijection between the two sets.

As for the second question, it appears this answer here shows us that for every group $G$ there is a topological space with that automorphism group. In particular it should happen when $G = \{e\}$, although of course we can find an example when our space has only one point.

However if you want the space to not be homeomorphic to any others then it's not possible because you could just "relabel" the points to get a different space, unless we take the empty topological space.

Asinomás
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  • Are the links the same? is that in both it sends me to the same post and in the 2nd I can't find the reference to the empty topological space – Haus May 18 '21 at 15:18
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    Fixed, sorry about that. – Asinomás May 18 '21 at 15:19
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    No problem, thanks for your contributions. – Haus May 18 '21 at 15:20
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    Oh my bad, incidentally I was also pointed to this source by someone in chat. It talks about spaces with trivial automorphism groups. https://mathoverflow.net/questions/188707/hausdorff-spaces-with-trivial-automorphism-group/188709#188709 which seem to be called rigid ( although I don't know if more conditions are needed to be called rigid). – Asinomás May 18 '21 at 15:29
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    Well, it seems that in the end there was a relationship with homogeneity – Haus May 18 '21 at 15:33
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    @Onir: Yes, a rigid space is one with a trivial automorphism group; that’s the only requirement, and there are lots of them. In fact, there are $2^\omega$ pairwise disjoint, pairwise non-homeomorphic, dense subspaces of $[0,1]$ that are rigid in a very strong way. – Brian M. Scott May 18 '21 at 19:03
  • Excuse me Professor @BrianM.Scott, do you have any reference to spaces that only support 2 homeomorphic spaces to him, or in general spaces that only support n homeomorphic spaces to him? – Haus Jul 17 '21 at 05:09
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    @Haus: Interesting question. You might start by taking a look at references $[3]$ and $[4]$ in this paper; it’s been $45$ years since I looked at them, so I don’t remember how helpful they might be, but at least they might give you a starting point and some terminology for searching. – Brian M. Scott Jul 17 '21 at 20:18
  • Thank you very much, i will check them – Haus Jul 17 '21 at 20:54
  • Hello, I have a question again. I continue to study this question, this by reviewing the article Rigid continuous and Topological Group-Pictures, by J. De Groot, at the beginning it mentions something like "a group of continuous potency", as much as I have looked for the definition I have not found it, you will have knowledge of what that refers to @BrianM.Scott? – Haus Oct 20 '21 at 02:38