Proving a Space is Normal

Topology · Axiom Academy

EXAMPLE Proving a Space is Normal Step-by-step proofs using separation axioms and key theorems Let be a metric space with the induced topology. Then is normal. Metric spaces are T₁ because for any two distinct points x ≠ y, we can separate them with open balls of radius d(x,y)/2. For a point x and a set A, we define the distance from x to A as the infimum of distances to all points in A. Construct Separating Open Sets We define U as all points closer to A than to B, and V as all points closer to B than to A. These are open by the continuity of the distance function. For any a ∈ A, we have d(a,A) = 0 while d(a,B) > 0 (since A and B are disjoint closed sets). Thus a ∈ U. Similarly B ⊆ V. If x ∈ U ∩ V, then d(x,A) < d(x,B) and d(x,B) < d(x,A), which is a contradiction. Therefore U ∩ V = ∅. Understanding the Construction Why is it crucial that A and B are both closed sets in the construction of U and V? Every Compact Hausdorff Space is Normal Let be a compact Hausdorff space. Then is normal. Hausdorff spaces are automatically T₁, since we can separate any two distinct points with disjoint open neighborhoods. For each b ∈ B, we use the Hausdorff property to find disjoint open sets separating A from b . This requires a key lemma about closed sets in Hausdorff spaces. The collection Vb : b ∈ B covers B. Since B is a closed subset of a compact space, B is compact. Extract a finite subcover.

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