Are continuous bijections between compact and topological spaces homeomorphisms?

Question

We know that continuous bijections between compact and Hausdorff spaces are homeomorphisms. I.e. if $(X,\tau_{X})$ is a compact topological space, $(Y,\tau_{Y})$ is a Hausdorff space and $f:(X,\tau_{X})\rightarrow(Y,\tau_{Y})$ is a continuous bijective function, then $f$ is a homeomorphism.

However, does this hold if we relax the assumption that $(Y,\tau_{Y})$ is a Hausdorff space, i.e. only a topological space? If not, provide a counterexample.

Can you find an example of a closed subset of a compact space that is not compact? — azif00, Mar 19 '21 at 16:09

score 3 · Accepted Answer · answered Mar 19 '21 at 16:22

3

Take $X$ to be a finite set ($|X|>1$) with discrete topology and $X'$ be the same set with indiscrete topology.

Consider $i : X \to X'$, the identity function.

Clearly, $X$ is compact (as it is finite). $i$ is clearly bijective and continuous (you cannot demand more open sets in the left space).

But, obviously, this is not a homeomorphism.

answered Mar 19 '21 at 16:22

MathIsNice1729

4,510

1

Similarly, take any compact space $X$ that is NOT indiscrete, and do the same thing. – Justin Young Mar 19 '21 at 16:24
@JustinYoung Yeah, this is just a very easy example. – MathIsNice1729 Mar 19 '21 at 16:26

Are continuous bijections between compact and topological spaces homeomorphisms?

1 Answers1