5. Separation Axioms

Separation Axioms

T0, T1, T2, and T3 Separation Axioms

T0 Separation Axiom: A topological space \( X \) satisfies the T0 separation axiom if for every pair of distinct points \( x, y \) in \( X \), there exists an open set containing exactly one of them.

T1 Separation Axiom: A topological space \( X \) satisfies the T1 separation axiom if for every pair of distinct points \( x, y \) in \( X \), there exist disjoint open sets containing \( x \) and \( y \) respectively.

T2 Separation Axiom (Hausdorff Property): A topological space \( X \) satisfies the T2 separation axiom if for every pair of distinct points \( x, y \) in \( X \), there exist disjoint open sets containing \( x \) and \( y \) respectively.

T3 Separation Axiom: A topological space \( X \) satisfies the T3 separation axiom if it satisfies both the T1 separation axiom and, additionally, for every closed set \( F \) and a point \( x \) not in \( F \), there exist disjoint open sets containing \( F \) and \( x \) respectively.

Regular and Normal Spaces

Regular Space: A topological space \( X \) is called regular if it satisfies the T3 separation axiom.

Normal Space: A topological space \( X \) is called normal if for any pair of disjoint closed sets \( A \) and \( B \) in \( X \), there exist disjoint open sets containing \( A \) and \( B \) respectively.

Urysohn's Lemma and Tietze Extension Theorem

Urysohn's Lemma: Let \( X \) be a normal topological space. For any two disjoint closed sets \( A \) and \( B \) in \( X \), there exists a continuous function \( f: X \rightarrow [0, 1] \) such that \( f(A) = \{0\} \) and \( f(B) = \{1\} \).

Tietze Extension Theorem: Let \( X \) be a normal topological space, and let \( A \) be a closed subset of \( X \). If \( f: A \rightarrow [a, b] \) is a continuous function, where \( [a, b] \) is a closed interval in \( \mathbb{R} \), then there exists a continuous function \( F: X \rightarrow [a, b] \) such that \( F|_A = f \), i.e., \( F(x) = f(x) \) for all \( x \) in \( A \).