How to interpret the theorem below? I find this theorem useful for proving other topological stuff (not important here), but generally, I still fail to grasp intuition about such claims.
To be particular: We claim that the mapping $f$ is extendable to closure of $A$ iff the inverse of $f$ separates any disjoint closed subsets of the "image space". Why should I look and say "yes, of course, this is equivalent"? Do you find this intuitive?
My thought process:
I think the inverse images $f^{-1}(B_1)$ and $f^{-1}(B_2)$ must be closed, since closedness is preserved by homeomorphisms (which are all continuous maps of topological spaces I guess). So I don´t get why we are saying they have disjoint closures (if they are closed, the closures are the same, or?) instead of simply they are disjoint. Also, I still don´t see why this is equivalent of $f$ having extension over the closure of $A$.
Source: The theorem is from Engelking - General Topology. Can be found online here.
To begin: the preimages are closed in $A$, most likely not in $X$. The point is that if $B_1$ and $B_2$ are closed and disjoint and yet $x$ is in the closure of both preimages then continuous extension becomes impossible, by continuity the image of $x$ under the extension should be in the intersection of $B_1$ and $B_2$ (which is empty). So this condition is definitely necessary. It is used in the proof of sufficiency to show that every point in $X\setminus A$ has at most one potential value: the intersection $\bigcap\{\overline{f[U\cap A]}:U$ a neighbourhood of $x\}$ is nonempty by compactness and consists of just one point by the condition on the closed sets.