A topological space \((X,\mathcal T)\) is called limit-point compact if every infinite subset \(A\) of \(X\) has a limit point. Show that every closed subset of a limit-point compact space is limit-point compact.
Solution
Let \(C\subseteq X\) be closed and \(A\subseteq C\) infinite.
1. \(A\) has a limit point \(x\in X\), and \(x\in C\).
\(A\) is an infinite subset of the limit-point compact space \(X\). A limit point of \(A\) lies in \(\overline A\subseteq C\), since \(C\) is closed.
2. Q.E.D.
Every neighborhood of \(x\) in \(C\) has the form \(U\cap C\) with \(U\) open in \(X\), and \(U\) meets \(A\setminus\{x\}\subseteq C\); so \(x\) is a limit point of \(A\) in \(C\). Hence \(C\) is limit-point compact.