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) 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.
 * 4) 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.
 * 5) Use Alexander's subbasis theorem to prove Tychonoff's theorem.
 * 6) 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.
 * 7) Let $$Z_1, \ldots, Z_n$$ 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 $$X$$.
 * 8) 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.
 * 9) Let $$X$$ be a noncompact connected topological space. Prove that its Alexandroff compactification $$X_\infty$$ is connected.
 * 1) Let $$X$$ be a noncompact connected topological space. Prove that its Alexandroff compactification $$X_\infty$$ is connected.