Let \(p: X \to Y\) be a closed continuous surjective map. Show that if \(X\) is normal, then so is \(Y\). [Hint: If \(U\) is an open set containing \(p^{-1}(\ts{y})\), show there is a neighborhood \(W\) of \(y\) such that \(p^{-1}(W) \subset U\).]
Solution
Goal: Prove that if \(X\) is a normal space and \(p: X \to Y\) is a closed continuous surjection, then \(Y\) is a normal space.
1. \(T_1\) separation axiom on \(Y\):Proof:
1.1. For any \(y \in Y\), the singleton \(\{y\} \subset Y\) is closed:- Since \(X\) is \(T_1\) (normal implies \(T_1\)), and \(p\) is continuous, the fiber \(p^{-1}(\{y\})\) is closed in \(X\).- Because \(p\) is a closed map and \(p\) is surjective, the image \(p(p^{-1}(\{y\})) = \{y\}\) is closed in \(Y\).
1.2. Hence \(Y\) satisfies the \(T_1\) axiom.
2. Pushing open saturated neighborhoods down via closed maps:Proof:
2.1. For any open set \(U \subseteq X\), define the saturated open set:\[W_U = Y \setminus p(X \setminus U).\]
2.2. Since \(X \setminus U\) is closed in \(X\) and \(p\) is a closed map, \(p(X \setminus U)\) is closed in \(Y\).
2.3. Therefore, \(W_U\) is an open subset of \(Y\).
2.4. For any \(y \in Y\), \(y \in W_U \iff y \notin p(X \setminus U) \iff p^{-1}(\{y\}) \subseteq U\).
2.5. Thus \(p^{-1}(W_U) \subseteq U\).
3. Separation of disjoint closed sets in \(Y\):Proof:
3.1. Let \(A\) and \(B\) be disjoint closed subsets of \(Y\).
3.2. By continuity of \(p\), the preimages \(p^{-1}(A)\) and \(p^{-1}(B)\) are closed subsets of \(X\).
3.3. Since \(A \cap B = \varnothing\), \(p^{-1}(A) \cap p^{-1}(B) = p^{-1}(A \cap B) = \varnothing\).
3.4. Since \(X\) is normal, there exist disjoint open sets \(U, V \subset X\) such that:\[p^{-1}(A) \subseteq U \quad \text{and} \quad p^{-1}(B) \subseteq V.\]
3.5. Applying the construction from 2, let \(W_U = Y \setminus p(X \setminus U)\) and \(W_V = Y \setminus p(X \setminus V)\).
3.6. Both \(W_U\) and \(W_V\) are open in \(Y\).
3.7. For any \(a \in A\), \(p^{-1}(\{a\}) \subseteq p^{-1}(A) \subseteq U\), so \(a \in W_U\); thus \(A \subseteq W_U\).
3.8. Symmetrically, \(B \subseteq W_V\).
3.9. Disjointness: \(p^{-1}(W_U \cap W_V) = p^{-1}(W_U) \cap p^{-1}(W_V) \subseteq U \cap V = \varnothing\).
3.10. Since \(p\) is surjective, the empty preimage implies \(W_U \cap W_V = \varnothing\).
4. Conclusion:\(A\) and \(B\) are separated by disjoint open sets \(W_U\) and \(W_V\), so \(Y\) is normal. Q.E.D.