Show that if \(X\) is regular, every pair of points of \(X\) have neighborhoods whose closures are disjoint.
Solution
Goal: Prove that in a regular (\(T_3\)) topological space \(X\), every pair of distinct points possesses open neighborhoods with disjoint closures (i.e. \(X\) is an Urysohn / \(T_{2\frac{1}{2}}\) space).
1. Separation of distinct points by disjoint open sets:Let \(x, y \in X\) be distinct points (\(x \neq y\)).There exist disjoint open sets \(U_0, V_0 \subseteq X\) such that \(x \in U_0\) and \(y \in V_0\).Proof:
1.1. In a regular space, one-point sets are closed (\(T_1\) axiom), so \(\{y\}\) is closed in \(X\).
1.2. Because \(x \neq y\), \(x \notin \{y\}\).
1.3. By regularity of \(X\), there exist disjoint open sets \(U_0\) containing \(x\) and \(V_0\) containing \(\{y\}\) such that \(U_0 \cap V_0 = \varnothing\).
2. Shrinking to open neighborhoods with closed containment:There exist open neighborhoods \(U \subseteq X\) of \(x\) and \(V \subseteq X\) of \(y\) such that \(\overline{U} \subseteq U_0\) and \(\overline{V} \subseteq V_0\).Proof:
2.1. By the standard characterization of regularity (Munkres Lemma 31.1), for any point \(p\) and open neighborhood \(W\) of \(p\), there exists an open neighborhood \(N\) of \(p\) with \(\overline{N} \subseteq W\).
2.2. Applying this to the point \(x\) and its open neighborhood \(U_0\), there exists an open neighborhood \(U\) of \(x\) such that \(\overline{U} \subseteq U_0\).
2.3. Applying this to the point \(y\) and its open neighborhood \(V_0\), there exists an open neighborhood \(V\) of \(y\) such that \(\overline{V} \subseteq V_0\).
3. Disjointness of the closures:Proof:
3.1. By 2, \(\overline{U} \subseteq U_0\) and \(\overline{V} \subseteq V_0\).
3.2. Therefore:\[\overline{U} \cap \overline{V} \subseteq U_0 \cap V_0.\]
3.3. By 1, \(U_0 \cap V_0 = \varnothing\), so \(\overline{U} \cap \overline{V} = \varnothing\).
4. Conclusion:\(U\) and \(V\) are open neighborhoods of \(x\) and \(y\) respectively with \(\overline{U} \cap \overline{V} = \varnothing\). Q.E.D.