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164 CHAPTER 7. METRIC SPACES AND TOPOLOGICAL SPACES

Definition 7.12.1 Let X be a nonempty set and suppose B ⊆P (X). Then B is a basisfor a topology if it satisfies the following axioms.

1.) Whenever p ∈ A∩B for A,B ∈B, it follows there exists C ∈B such that p ∈C ⊆A∩B.

2.) ∪B = X.Then a subset, U, of X is an open set if for every point, x ∈U, there exists B ∈B such

that x ∈ B ⊆U. Thus the open sets are exactly those which can be obtained as a union ofsets of B. Denote these subsets of X by the symbol τ and refer to τ as the topology or theset of open sets.

Note that this is simply the analog of saying a set is open exactly when every point isan interior point.

Proposition 7.12.2 Let X be a set and let B be a basis for a topology as defined aboveand let τ be the set of open sets determined by B. Then

/0 ∈ τ, X ∈ τ, (7.12.12)

If C ⊆ τ, then ∪C ∈ τ (7.12.13)

If A,B ∈ τ, then A∩B ∈ τ. (7.12.14)

Proof: If p ∈ /0 then there exists B ∈B such that p ∈ B⊆ /0 because there are no pointsin /0. Therefore, /0 ∈ τ . Now if p ∈ X , then by part 2.) of Definition 7.12.1 p ∈ B ⊆ X forsome B ∈B and so X ∈ τ .

If C ⊆ τ, and if p ∈ ∪C , then there exists a set, B ∈ C such that p ∈ B. However, B isitself a union of sets from B and so there exists C ∈B such that p ∈C ⊆ B ⊆ ∪C . Thisverifies 7.12.13.

Finally, if A,B∈ τ and p∈A∩B, then since A and B are themselves unions of sets of B,it follows there exists A1,B1 ∈B such that A1 ⊆ A,B1 ⊆ B, and p ∈ A1∩B1. Therefore, by1.) of Definition 7.12.1 there exists C ∈B such that p ∈C⊆ A1∩B1 ⊆ A∩B, showing thatA∩B ∈ τ as claimed. Of course if A∩B = /0, then A∩B ∈ τ . This proves the proposition.

Definition 7.12.3 A set X together with such a collection of its subsets satisfying 7.12.12-7.12.14 is called a topological space. τ is called the topology or set of open sets of X.

Definition 7.12.4 A topological space is said to be Hausdorff if whenever p and q aredistinct points of X, there exist disjoint open sets U,V such that p ∈U, q ∈ V . In otherwords points can be separated with open sets.

Hausdorff

·p

U

·q

V