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.