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 [...].