Normal Spaces

Topology · Axiom Academy

The strongest separation property: keeping closed sets completely apart Step 1: What Makes a Space Normal? We've seen how T 1 and Hausdorff spaces separate points. But what if we want to separate entire closed sets from each other? This leads us to the strongest separation axiom in the T-hierarchy: normal spaces . A topological space is called normal (or satisfies the T 4 axiom when combined with T 1 ) if for any two disjoint closed sets and , there exist disjoint open sets and such that: Just like with other separation axioms, we combine normality with the T 1 property to get the T 4 axiom . A topological space is called a T 4 space if it is both: Normal : Disjoint closed sets can be separated by disjoint open sets T 1 : All singletons are closed The T 1 condition ensures that the space has enough closed sets to make normality meaningful. Without T 1 , normality becomes a weaker condition. Step 3: Example - The Real Line is Normal Let's verify that with the standard topology is a normal space. Claim: The real line ℝ with the standard topology is normal. Proof sketch: Let and be two disjoint closed sets in ℝ. For each point , since is closed and doesn't contain , we can find a distance such that the open interval of radius around doesn't touch . Similarly, for each , find such that the radius interval around doesn't touch . Then are disjoint open sets containing respectively. Step 4: Every Metric Space is Normal

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