Intuitive difference between a continuous map and a homeomorphism

Question

I know the formal definitions of both continuous map and a homeomorphism between two spaces.

If two spaces are homeomorphic intuitively they can be thought of spaces which can converted to each other through streching or contracting action.

But what is not clear to me is what is the intuitive idea of a continuous map between two spaces?

Why does the additional property of two way continuity required for spaces to be homeomorphic?

So far I have understood the intuitive idea behind continuous maps like retraction etc. but for a general continuous map it is still a mystery.

I felt it was very important for me to understand this in order to appreciate the difference between simplical and singular homology.

Answer 1

The intuitive idea behind a continuous map is a "map that keeps nearby points nearby".

Take a look at the following simple theorem/exercise:

If $f:X\to Y$ is continuous and $A\subset X$ is connected then $f(A) \subset Y$ is connected.

Thus, at least, a continuous function cannot separate a connected set. Now, this "global" statement can also be stated "locally": For simplicity, suppose $X$ is a locally-connected space. Take some $x\in X$ and take smaller and smaller connected neighborhoods $A$ of $x$. Then their images are connected neighborhoods of $f(x)$, also becoming smaller and smaller. But no matter how small, all points in $f(A)$ must be connected, through this neighborhood, to $f(x)$, so the nearby points (from $A$) translate to nearby points (from $f(A)$).

Since you mentioned homology, let me also answer more generally. Take a categorical point of view: The category of topological spaces. The continuous functions are simply the morphisms of this category - these are precisely the functions that preserve topological structure.

In general, take a category of sets with an additional structure, such as topological spaces, or groups, or rings, or vector spaces, etc. Taking any two sets $X,Y$ in this category, there are many functions $f:X\to Y$ but not all of them are worthy of being morphisms - some of them are just set functions and don't preserve the additional structure. In the case of topology, the structure is the declaration of open sets, and the preservation of the structure turns out to be that the preimage of an open set is open, which is precisely the definition of a continuous function.

When I say the structure is preserved, I don't mean that the image looks like the origin. I mean that the image looks like a topological space - it has topological structure.

Whatever the category, once morphisms (topology - continuous functions, groups - homomorphisms, vector spaces - linear transformations, etc.) are known, the definition of an isomorphism (in topology - homeomorphism) is always more or less the same:

A morphism is an isomorphism if it is invertible and its inverse is also a morphism.

Then not only does it preserve (topological) structure, but the two structures are identical.