General Topology/Compact spaces

Note that the composition of proper maps is proper.

Conversely, we have:

Often, $X\subset Y$ and $f$ is the inclusion.

Exercises

1. Let $S$ be a set, $X$ a topological space, and $f:S\to X$ a function. Then $f(S)$ is a compact subset of $X$ if and only if there exists a topology on $S$ which makes $S$ into a compact topological space.
2. Let $X$ be a set with two topologies ${\tau }_1$ and ${\tau }_2$, with respect to which $X$ is compact. Prove that also, $X$ is compact with respect to the topologies ${\tau }_1\cap {\tau }_2$ and ${\tau }_1\vee {\tau }_2$, where the latter shall denote the least upper bound topology of ${\tau }_1$ and ${\tau }_2$, borrowing notation from lattice theory.
3. 1. Let $X,Y$ be topological spaces and let $K\subseteq X$ and $L\subseteq Y$ be compact sets. Prove that $K\times L$ is a compact subset of $X\times Y$, where the latter is given the product topology.
   2. Let $X,Y$ be Hausdorff spaces and suppose that $f:X\to Y$ and $g:Y\to Y$ are proper, continuous functions. On the condition of the axiom of choice, prove that $f\times g:X\times Y\to Y\times Y$ is proper. Hint: Prove first that it suffices to show that preimages of products of compact sets of $Y$ are compact.
4. Use Alexander's subbasis theorem to prove Tychonoff's theorem.
5. Prove that if $X$ is a compact space and $A\subseteq X$ is discrete with respect to the subspace topology, then $A$ is a finite set.
6. Let $Z_1,\dots ,Z_n$ be compact spaces, $X$ a set and for $k\in [n]$, let $f_k:Z_k\to X$ be a function. Suppose that $X$ carries the final topology by the $f_k$ ($k\in [n]$). Prove that $X$ is compact if and only if $\bigcup _{k=1}^n{f}_k(Z_k)$ is cofinite in $X$.
7. Let $X,Y$ be topological spaces, where $X$ is compact and $Y$ is Hausdorff, and let $f:X\to Y$ be a continuous bijection. On the condition of the axiom of choice, prove that $X$ is Hausdorff and $Y$ is compact.
8. Let $X$ be a noncompact connected topological space. Prove that its Alexandroff compactification $X_{\mathrm{\infty }}$ is connected.