Regular Spaces

Topology · Axiom Academy

Unit 6 - Separation Axioms: When points and closed sets can be separated Step 1: The Need for Regularity So far, we've studied the T 0 and T 1 separation axioms, which help us distinguish points from each other. But what about separating a point from a closed set that doesn't contain it? The concept of regularity addresses this more sophisticated separation requirement. It ensures that points and closed sets can be "kept apart" using disjoint open neighborhoods. Step 2: Definition of Regular Spaces A topological space is called regular if for every closed set and every point , there exist disjoint open sets and such that and . In other words, a space is regular if we can always find open neighborhoods around a point and around a closed set (not containing that point) that don't overlap at all. A topological space is called T 3 (or regular Hausdorff ) if it is both: Regular (satisfies the separation condition above) T 1 (singleton sets are closed) The T 3 axiom combines regularity with the T 1 property. This is important because regularity alone doesn't prevent "trivial" spaces where many points are indistinguishable. Points can be separated from closed sets not containing them using disjoint open sets. Regular AND every singleton set x is closed. Step 4: Examples of Regular Spaces Example 1: Metric Spaces are T 3 Every metric space is T 3 . Given a closed set C and a point , let . Since C is closed and x is not in C, we have .

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