State the definition of what it means for a topological space \(X\) to be regular.
Prove that if \(X\) and \(Y\) are regular, then the space \(X\times Y\) with the product topology is regular.
Solution
1. (a) A space \(X\) is regular if it is \(T_1\) and, whenever \(x\in X\) and \(C\subseteq X\) is closed with \(x\notin C\), there are disjoint open sets \(U\ni x\) and \(W\supseteq C\).
2. A \(T_1\) space \(X\) is regular if and only if for every \(x\in X\) and every open \(U\ni x\) there is an open \(V\ni x\) with \(\overline V\subseteq U\).
If \(X\) is regular and \(x\in U\), separate \(x\) from \(X\setminus U\) by disjoint open \(V\ni x\) and \(W\supseteq X\setminus U\); then \(\overline V\subseteq X\setminus W\subseteq U\). Conversely, given \(x\notin C\) with \(C\) closed, choose \(V\ni x\) with \(\overline V\subseteq X\setminus C\); then \(V\) and \(X\setminus\overline V\) separate \(x\) and \(C\).
3. (b) If \(X\) and \(Y\) are regular, then \(X\times Y\) is regular.
Singletons are closed in \(X\) and \(Y\), so \(\{(x,y)\}=\{x\}\times\{y\}\) is closed in \(X\times Y\), which is therefore \(T_1\). Let \(O\) be an open neighborhood of \((x,y)\), and choose open \(U\ni x\) and \(V\ni y\) with \(U\times V\subseteq O\). By step 2, there are open \(U'\ni x\) and \(V'\ni y\) with \(\overline{U'}\subseteq U\) and \(\overline{V'}\subseteq V\). Since \(\overline{U'\times V'}=\overline{U'}\times\overline{V'}\), we get \(\overline{U'\times V'}\subseteq U\times V\subseteq O\), and step 2 shows that \(X\times Y\) is regular.
4. Q.E.D.
Step 1 answers (a), and step 3 proves (b).