7.12. GENERAL TOPOLOGICAL SPACES 165

Definition 7.12.5 A subset of a topological space is said to be closed if its complement isopen. Let p be a point of X and let E ⊆ X. Then p is said to be a limit point of E if everyopen set containing p contains a point of E distinct from p.

Note that if the topological space is Hausdorff, then this definition is equivalent torequiring that every open set containing p contains infinitely many points from E. Why?

Theorem 7.12.6 A subset, E, of X is closed if and only if it contains all its limit points.

Proof: Suppose first that E is closed and let x be a limit point of E. Is x ∈ E? If x /∈ E,then EC is an open set containing x which contains no points of E, a contradiction. Thusx ∈ E.

Now suppose E contains all its limit points. Is the complement of E open? If x ∈ EC,then x is not a limit point of E because E has all its limit points and so there exists an openset, U containing x such that U contains no point of E other than x. Since x /∈ E, it followsthat x ∈U ⊆ EC which implies EC is an open set because this shows EC is the union ofopen sets.

Theorem 7.12.7 If (X ,τ) is a Hausdorff space and if p ∈ X, then {p} is a closed set.

Proof: If x ̸= p, there exist open sets U and V such that x ∈U, p ∈ V and U ∩V = /0.Therefore, {p}C is an open set so {p} is closed.

Note that the Hausdorff axiom was stronger than needed in order to draw the conclusionof the last theorem. In fact it would have been enough to assume that if x ̸= y, then thereexists an open set containing x which does not intersect y.

Definition 7.12.8 A topological space (X ,τ) is said to be regular if whenever C is a closedset and p is a point not in C, there exist disjoint open sets U and V such that p ∈U, C⊆V .Thus a closed set can be separated from a point not in the closed set by two disjoint opensets.

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Definition 7.12.9 The topological space, (X ,τ) is said to be normal if whenever C and Kare disjoint closed sets, there exist disjoint open sets U and V such that C ⊆U, K ⊆ V .Thus any two disjoint closed sets can be separated with open sets.

Normal

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