2.6_基与子基(点集拓扑)详解.ppt

We want to invent ways to generate a topology on a set. This is the purpose of the present section, and here is the first of two ways we will do this. Base and Subbase for a Topology Definition 2.6.1. A collection of open sets in a topological space X is called a basis for the topology if every open set in X is a union of sets in . 问 题 任何一个拓扑空间是否都存在基？ 拓扑空间的基是否存一？ Example1 (a) If (X, T ) is a topological space, then T is a base for itself. (b) If (X, d) is a metric space, then B = {B(x; r) : x ∈ X, r 0} is a base for the topology on X. (c) If (X, d) is a metric space, then B = {B(x; r) : x ∈ X, r 0 and r ∈ Q} is a base for the topology on X. Thus we see that a base is not unique. Theorem 2.6.2 Let (X, T ) be a topological space. A family B ? T is called a base for (X, T ) if and only if whenever x ∈ G ∈ T , ?B ∈ B such that x ∈ B ? G. Lemma Theorem 2.6.3 Example2 Example3 Definition 2.6.2. Example 8 in Example 7. Theorem 2.6.5 The following are equivalent for a map f : X → Y between topological spaces.