Chapter 19

Hausdorff Spaces and Measures19.1 General Topological Spaces

It turns out that metric spaces are not sufficiently general for some applications. This sec-tion is a brief introduction to general topology. In making this generalization, the propertiesof balls in a metric space are stated as axioms for a subset of the power set of a given set X .This subset of the power set P (X) (set of all subsets) will be known as a basis for a topol-ogy. The properties of balls which are of importance are that the intersection of finitelymany is the union of balls and that the union of all of them give the whole space. Recallthat with a metric space, an open set was just one in which every point was an interiorpoint. This simply meant that every point is contained in a ball which is contained in thegiven set. All that is being done here is to make these simple properties into axioms.

Definition 19.1.1 Let X be a nonempty set and suppose B ⊆P (X). Then B is abasis for 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 19.1.2 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, (19.1)

If C ⊆ τ, then ∪C ∈ τ (19.2)

If A,B ∈ τ, then A∩B ∈ τ. (19.3)

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 19.1.1 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 19.2.

Finally, if A,B ∈ τ and p ∈ A∩B, then since A and B are themselves unions of sets ofB, it follows there exists A1,B1 ∈B such that A1 ⊆ A,B1 ⊆ B, and p∈ A1∩B1. Therefore,by 1.) of Definition 19.1.1 there exists C ∈B such that p ∈C ⊆ A1∩B1 ⊆ A∩B, showingthat A∩B ∈ τ as claimed. Of course from the above, if A∩B = /0, then A∩B ∈ τ . ■

Definition 19.1.3 A set X together with such a collection of its subsets satisfying19.1-19.3 is called a topological space. τ is called the topology or set of open sets of X.

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