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Let $X$ and $Y$ be topological spaces, $X$ compact, $f : X \to Y$ continuous. Then the preimage of each compact subset of $Y$ is compact.

With the stipulation that $X$ and $Y$ are metric spaces, this is a theorem in Pugh's Real Mathematical Analysis. The proof uses sequential compactness. Is this theorem true in general (i.e. can it be proved with covering compactness alone)?

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Are your spaces Hausdorff? – egreg Feb 14 '14 at 22:02
It holds if $Y$ is Hausdorff. If it isn't, generally not. – Daniel Fischer Feb 14 '14 at 22:03
Maybe. Does it matter? It's not a homework assignment or anything, some friends and I were just wondering. – Brian Bi Feb 14 '14 at 22:03
Oh okay, can you post a counterexample? – Brian Bi Feb 14 '14 at 22:03
@BrianBi If $Y$ is Hausdorff, a compact subset is closed, so the inverse image is closed in $X$, hence compact. If $Y$ is indiscrete… – egreg Feb 14 '14 at 22:04
up vote 11 down vote accepted

The claim is true if $Y$ is Hausdorff: if $C\subseteq Y$ is compact, then it is closed; therefore $f^{-1}(C)$ is closed in $X,$ hence compact.

For a counterexample, take $Y=X$ with the indiscrete topology and $f$ the identity map. Then every subset of $Y$ is compact, which can be easily arranged for $X$ not to.

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This is not true in general.

Let $X=Y=[0,1]$. Take $X$ with the usual topology. For $Y$, take the topology $$\tau=\left\{\varnothing,Y,(1/2,1]\right\}.$$ Then $id:x\in X\mapsto x\in Y$ is continuous, but $(1/2,1]=id^{-1}(1/2,1]$ is not compact, although $(1/2,1]$ is compact in $Y$.

On the other hand, if $Y$ is Hausdorff, then every compact of $Y$ is closed, so the inverse image of compact sets is closed, hence compact.

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A map $f:X\to Y$ is called proper if the preimage of every compact subset is compact. It is called closed if the image of every closed subset is closed.

If $X$ is a compact space and $Y$ is a Hausdorff space, then every continuous $f:X\to Y$ is closed and proper.

Here are some examples where $f$ is not proper:

  • With $X$ compact: Let $X=[0,1]$ and $f=\text{Id}:(X,\tau)\to(X,\sigma)$ where $\tau$ is the Euclidean topology and $\sigma$ is the cofinite topology.
  • With $Y$ Hausdorff: Let $f:\Bbb R\to\{*\}$ be the constant map.
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