Show that a closed subset of a Hausdorff space need not be compact.
Solution
1. Take \(X=\mathbb R\) with its usual topology and \(A=\mathbb R\).
The real line is Hausdorff, and \(A=X\) is closed in \(X\).
2. The set \(A\) is not compact.
The open cover \(\{(-n,n):n\ge1\}\) has no finite subcover, since every finite subfamily is contained in \((-N,N)\) for some \(N\).
3. Thus a closed subset of a Hausdorff space need not be compact.
The example in <1>1–<1>2 has all the required properties.