Show that if \(X\) is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint.
Solution
1. Initial separation of disjoint closed sets:
1.1. Let \(A\) and \(B\) be disjoint closed subsets of a normal space \(X\) (\(A \cap B = \emptyset\)).
setup.
1.2. By the definition of normality, there exist disjoint open sets \(U, V \subseteq X\) such that:\[ A \subseteq U, \quad B \subseteq V, \quad U \cap V = \emptyset. \]
definition of a normal topological space.
1.3. Since \(U \cap V = \emptyset\), we have \(U \subseteq X \setminus V\).Since \(V\) is open, \(X \setminus V\) is closed, so taking closures yields:\[ \overline{U} \subseteq X \setminus V \implies \overline{U} \cap V = \emptyset. \]
closure of a subset of a closed set is contained in the closed set.
2. Strengthening to separated closures:
2.1. Consider the closed set \(A\) and the closed set \(X \setminus U\).Since \(A \subseteq U\), \(A \cap (X \setminus U) = \emptyset\).
set complement.
2.2. By normality, applying the open neighborhood lemma to \(A\) and \(X \setminus U\), there exists an open set \(W \subseteq X\) such that:\[ A \subseteq W \quad \text{and} \quad \overline{W} \subseteq U. \]
characterization of normality (\(A \subseteq U\) open \(\implies \exists W\) open with \(A \subseteq W \subseteq \overline{W} \subseteq U\)).
2.3. Symmetrically, since \(B\) is closed and \(B \subseteq V\), there exists an open set \(G \subseteq X\) such that:\[ B \subseteq G \quad \text{and} \quad \overline{G} \subseteq V. \]
normality applied to \(B \subseteq V\).
3. Prove that the closures of \(W\) and \(G\) are disjoint:
3.1. By construction, \(\overline{W} \subseteq U\) and \(\overline{G} \subseteq V\).
2 steps 2 and 3.
3.2. Therefore:\[ \overline{W} \cap \overline{G} \subseteq U \cap V = \emptyset. \]Thus \(\overline{W} \cap \overline{G} = \emptyset\).
subset of the empty set is empty.
4. Conclusion:\(W\) and \(G\) are open neighborhoods of \(A\) and \(B\) respectively with \(\overline{W} \cap \overline{G} = \emptyset\). Q.E.D.
2 and 3.