Inverse image of a compact set is compact 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)?
 A: 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.
A: 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.

A: 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.
A: Disclaimer:  This answer is intended to compliment the above, and thought it is not fully generally, but is has often been helpful in my experience...  
If $Y$ is $\mathbb{R}$ then its sufficient for $f$ to be coercive.  When $X$ is a topological vector space, the coercivity condition reduces to the requirement that
$$
\lim_{\|x\|\to \infty} f(x)=\infty,
$$
which is typically easy to check.  
