8.4 Compactness[2BF]

Definition 252

[0J3] A subset \(K⊆ X\) is compact 1 if, from every family of open sets \((A_ i)_{i∈ I}\) whose union \(⋃_{i∈ I}A_ i\) covers \(K\), we can choose a finite number \(J⊂ I\) of open set whose union \(⋃_{i∈ J}A_ i\) covers \(K\).

If you formulate these exercises in metric spaces, you can use the theorem 299 to deal with compact sets.

E252

[0J4] Suppose the topological space is compact. Show that every closed subset is compact.

E252

[0J5] Suppose the topological space is \(T_ 2\) (see Definition 245). Prove that every compact subset is closed.

E252

[0J6] Topics:compact sets.Prerequisites:2. Note:For the real case, see 5. For the case of metric spaces, see 9..

Let \((X,𝜏)\) be a \(T_ 2\) topological space and let \(A_ n⊆ X\) be compact nonempty subsets such that \(A_{n+1}⊆ A_ n\): then \(⋂_{n∈ℕ} A_ n≠ ∅\).

What happens if the space is not \(T_ 2\)? Hidden solution: [UNACCESSIBLE UUID ’0J7’]

E252

[0J8] Prerequisites:2.Let \((X,𝜏)\) and \((Y,𝜎)\) be topological spaces, with \(X\) compact and \(Y\) \(T_ 2\). Let \(f:X→ Y \) be continuous and injective; show that \(f\) is a homeomorphism between \(X\) and its image \(f(X)\).

Hidden solution: [UNACCESSIBLE UUID ’0J9’]

E252

[0JB]Prerequisites:2.Show that the extended line (the topological space shown in 2) is compact. Hidden solution: [UNACCESSIBLE UUID ’0JC’]

E252

[0JD]Prerequisites:3.Show that the compacted line (the topological space shown in 3) is compact.

See also the exercise 5 for a characterization of compact sets by nets.

  1. The definition shows that the empty set is compact. Some texts however explicitly exclude this case.