Why are homeomorphic spaces regarded as similar? Doing topology this morning, I asked myself a very simple question but I wasn’t able to answer it properly anymore. It was to know why homeomorphic spaces are regarded as similar ? On examples, we see that they resemble very much, but I do not see anymore why the important properties in topology are stable under homeomorphism, what are those properties and why.
Thanks in advance:)
 A: Informal answer.
If $X$ and $Y$ are topological spaces and there is a homeomorphism $X\to Y$ then actually you can interpret the underlying set of $Y$ as the set $\{\hat x\mid x\in X\}$ where $x\mapsto\hat x$ is the prescription of the homeomorphism.
But what's the difference between $X=\{x\mid x\in X\}$ and $\{\hat x\mid x\in X\}$?...
The topologies are unaffected in the sense that:$$\{x\mid x\in A\}\in\tau_X\iff\{\hat x\mid x\in A\}\in\tau_Y$$
Apparantly you can think of $Y$ as $X$ itself except that every element name $x$ is interchanged for element name $\hat x$ (i.e. $x$ just placed a hat on its head).
I can imagine that this is regarded as "too superficial" because element $\hat x$ might differ from $x$ not only by its hat. Nevertheless looking at it this way helps me a lot to understand that $X$ and $Y$ are "almost the same". Not only if it comes to topologies but also other mathematical structures that are - what we call - isomorphic.
Essential is:
The structure will not be affected if the elements only get a different nature.
A: Can you regard the sets $A=\{1,2,3\}$ and $B=\{James,Bill\}$ as similar? Clearly not.
What about the sets $A=\{1,2,3\}$ and $B=\{James,Bill,Charles\}$? You can think that is is the case, because you can define a map $f$ such that:
$$1 \to James$$
$$2 \to Bill$$
$$3 \to Charles$$
and then they are the same with the only difference of "notation".
But, what if we put "more structure" on the sets? For example, suppose that you fix in $A$ the natural order 1,2,3 and in $B$ you order these people by age: James is older that Bill and Bill and Charles and twin brothers.
With these structures, you cannot say that the sets are "similar". But if it happens that Bill is older than Charles then you can. So in any conversation (about ages) you could substitute the names of these people by the corresponding numbers and the meaning wouldn't be affected.
With topology the same happens, only that now the structure is more complicated. You are distinguishing priviledged subsets (the open sets) that give us all the topological characteristics of the space. If you have a homeomorphism between the topological spaces, all the topological properties of $B$ (all the conversations that you can have about $B$ with respect to topology) are the same than that of $A$
A: It's all about choosing a lens through which we see the world.
Imagine you're wearing glasses with a strong red filter on the lenses. With those glasses on, it's possible to look at two balls with different colors and see them as identical, because they are matching in red levels. Switch to blue filter glasses and they may be entirely different, but now there's a separate pair of balls which look identical that didn't before. So, we have things that are "red isomorphic", and those that are "blue isomorphic", and they're different from each other. The lens determined what looked the same to us.
In topology, that lens is, well, the topology. With our topology glasses on, we can only see the structure of open subsets on a set. Because continuous functions preserve that topological structure, any two spaces related by a reversible continuous function will look the same in our topology glasses. To use the common example, put a donut and a coffee mug on the table and they will look identical in the glasses, even though they are obviously different in "reality".
The benefit of this approach is that any topological property which we can show for one example will hold for any space which is topologically the same. Any theorem on donuts works on coffee mugs; any theorem on circles works on the batman logo. The ability to study a simple object in place of a complex one makes our job that much easier.
