What means that two manifolds have "the same topology"?

Question

I know this is a very basic question, but let me be more specific. Suppose that, for definiteness, $M$ and $N$ are differentiable manifolds. What means that they have the same topology? Does this mean that there is an homeomorphism $f\colon M\to N$? I know that such homeomorphism implies that $M$ and $N$ can be equipped with "equivalent" topologies but, is there a map more general than an homeomorphism that "preserves" the topologies? For example, if there exist a map $h\colon M\to N$ such that $h$ is an homotopy equivalence, then $M$ and $N$ have the same topology?

My final question is: Suppose that for $M$ you can find an atlas with a single chart, but for $N$ the minimal atlas has two charts. Can they ever have the same topology? In that case if they have the same topology it cannot be because there is an homeomorphism $f\colon M\to N$, because then you could use it to find an atlas with only one chart for $N$.

Thanks.

Answer 1

Saying that $X$ and $Y$ have the same topology means that $X$ and $Y$ are homeomorphic to one another.

If $M$ admits a one element atlas, then $M$ is an open subset of $\mathbb{R}^n$ for some $n$. If $N$ does not admit such an atlas, it cannot be homeomorphic to $M$.

Answer 2

As a topology is a subset of the set of subsets of a given set, it is definied on this given set, so it either means that $N = M$ and that the two different topologies happen to be the same, of that there is an "isorphism" between $N$ and $M$ is some category, that induces an homeomorphism between them, so that one can identify $M$ and $N$ and their topologies.

Two topologicals spaces homotopically equivalent are not necessarily homeomorphic.

I you have an atlas of $M$ and if $M$ is homeomorphic to $N$, you can transport $M$'s atlas on $N$.

Answer 3

topology of X is open sets in X,so if f preserve topology and U is open we must have f(U) is open,and if V is open in Y f^{-1}(Y) is open and this is means f is homomorphism