Under what circumstances does continuity of a function imply continuity of its inverse?

Question

I was working on a few proofs about topological equivalence, and I realized that things would be a lot easier if I could form some shortcut that exempted me from proving the continuity of both a function mapping two topological spaces and its inverse. So I conjectured the following, and attempted to prove it:

If a function $f: A\to B$ is bijective and continuous, then its inverse $f^{-1}:B\to A$ is also continuous.

However, that was false, which surprised me. I found a counterexample at this previously asked math SE question. In the given counterexample, the set $A$ from which $f$ was mapping was split into two separate intervals.

My question is this: under what conditions is my conjecture true? What added property must $f$, $A$, and $B$ have for continuity to imply bicontinuity? My intuition tells me that most functions have this property, but I don't know how to tell when.

Answer 1

You add that $f$ is an open map.

Answer 2

One useful sufficient condition is that if the domain is compact and the range is Hausdorff then $f$ is bicontinuous.

Another sufficient condition with a weaker constraint on the domain says that if the domain is locally compact, and the range is Hausdorff, and $f$ is a proper map, then $f$ is bicontinuous.

You can find the proofs of these in most topology books, for example Munkres "Topology".

Answer 3

A map $f:A\to B$ is continuous if $f^{-1}(V)$ is open for every open set $V\subset B$. To ensure that $f^{-1}$ is continuous you need the condition that only the open sets of $B$ meet this property. That is, if there is a subset $V'\subset B$ that is not open, but $f^{-1}(V')$ is open in $A$, then $f^{-1}$ is not continuous.

Intuitively, this means that $B$ has a coarser topology than $A$. That is, there are open sets in $A$ that correspond to non-open sets of $B$.

For an "extreme" example, take a set $X$ with more than one point and consider in it the discrete topology $\tau$ (every subset is open) and the coarse topolgy $\tau'$ (only $\emptyset$ and $X$ are open). Let $f$ be the identity map. Then $f:(X,\tau)\to (X,\tau')$ is continuous, but its inverse is not.