Continuous functions and preservation of topological structure. In some book I read "continuous fuctions between topological spaces preserve their topological structure" and they say that this is similar to the case when homomorphims perserve algebraic structures. But thinking about it, it seems to me that this structure preservation is made backwards.
For example, given $f:X \Rightarrow Y$, both $X$ and $Y$ topological spaces, it is true that the inverse image of finite intersection of open sets in $Y$ will be open in $X$. But this not happen with direct images. I might conclude that topological structure of $Y$ is preserved when the inverse images of $f$ send it to $X$.
It that true? Thanks in advance.
 A: It’s complicated. Continuous functions do preserve many topological characteristics, among them compactness, connectedness, separability, and limits of convergent sequences. However, there are many topological characteristics that they do not necessarily preserve, even things as basic and important as being Hausdorff. This really is somewhat like the behavior of homomorphisms in algebra, which preserve some structure of the domain but can also lose a great of that structure. And just as isomorphisms are especially nice homomorphisms, homeomorphisms (which are clearly the topological analogue of algebraic isomorphisms) are especially nice continuous functions. But the analogy has some real limitations. For instance, a bijective homomorphism is an isomorphism, but a continuous bijection need not be a homeomorphism. I’d say that the analogy is qualitatively reasonable but cannot be pushed very far at a more detailed level.
It is true that continuous functions inversely preserve some topological characteristics, but note that if $f:X\to Y$ is a continuous function, the inverse map in question is not from $Y$ to $X$: it’s the function $\wp(Y)\to\wp(X):A\mapsto f^{-1}[A]$. This map does take open sets to open sets and closed sets to closed sets, for instance. But overall it transfers few topological properties from $Y$ to $X$, even when $X$ and $Y$ have the same underlying set: no matter what the topology on $Y$, $f$ will be continuous if $X$ has the discrete topology. You might note that something similar happens in the algebraic case: if $h:G\to H$ is a homomorphism, and $K$ is a subgroup of $H$, then $h^{-1}[K]$ is a subgroup of $G$. That is, homomorphisms, like continuous functions, inversely preserve some ‘nice’, structurally important subsets — but it’s still the maps themselves that we consider structure-preserving.
A: In general, the inverse of a continuous function $f : X \rightarrow Y$ is not a function, so the claim that

I might conclude that topological structure of $Y$ is preserved when the inverse images of $f$ send it to $X$.

is not well-defined (or at least it's not obvious what well-defined statement you're trying to make).
However, the original statement is accurate. The definition of continuity, $O$ open in $Y \implies f^{-1}(O)$ open in $X$ turns out to be exactly what you need to prove that various properties are invariant under continuous functions. You can just take a look at examples like the proof for connectedness. There are many others.
