Show that a finite union of compact subspaces of \(X\) is compact.
Solution
Let \(K_1,\dots,K_n\) be compact subspaces of \(X\), and let \(\mathcal U\) be an open cover of \(K_1\cup\cdots\cup K_n\). For each \(i\), the family \(\mathcal U\) covers \(K_i\), so compactness gives a finite subfamily \(\mathcal U_i\subset\mathcal U\) covering \(K_i\). Then \[ \mathcal U_1\cup\cdots\cup\mathcal U_n \] is finite and covers the union. Hence every finite union of compact subspaces is compact.