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Math and science::Topology

Basis

Instead of specifying all possible open sets of a topology, it is convenient to be able to specify the topology in terms of a smaller set. Analogously, for metric spaces, the set of open balls could be used to describe a metric space instead of directly specifying the arbitrary open sets. For topological spaces, a basis carries out this role (plural: bases).

Basis, definition

Let \( X \) be a topological space. A basis for \( X \) is a collection \( \mathscr{B} \) of open subsets of \( X \) such that every open subset of \( X \) is [...].

Two consequence of this definition are:

  • Every element of \( X \) [has a something that is something].
  • the intersection of [two somethings is a something].

A bit more formally, these two consequences translate to:

Lemma.

Let \( X \) be a topological space, and let \( \mathscr{B} \) be a basis for \( X \). Then:

  1. For each \( x \in X \), there is [...].
  2. If \( x \in B_1 \cap B_2 \) where \( B_1 \) and \( B_2 \) are basis elements, then [...].