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 $B$ of open subsets of $X$ such that every open subset of $X$ is a union of sets in $B$ .

Two consequence of this definition are:

Every element of $X$ has at least one neighbourhood that is a basis element.
the intersection of any two basis elements must be a basis element.

A bit more formally, these two consequences translate to:

Lemma.

Let $X$ be a topological space, and let $B$ be a basis for $X$ . Then:

For each $x \in X$ , there is at least one basis element $B$ containing $x$ .
If $x \in B_{1} \cap B_{2}$ where $B_{1}$ and $B_{2}$ are basis elements, then $x \in B_{3}$ for some basis element $B_{3} \subseteq B_{1} \cap B_{2}$ .

The choice of definition and lemma

The wording of this lemma is presented in line with Munkres's presentation. Munkres, however, presents this lemma as the definition of a basis; he then proceeds to derive the content of the definition that I've presented above. Leinster proceeds in the opposite order, as presented here. I think that the motivation of a basis is clearer from Leinster's definition, whereas the lemma presented by Leinster (definition for Munkres) is more useful to work with. I followed Munkres's wording for the lemma as I feel it encourages thinking in terms of individual elements of $X$ which so far has been useful in proofs.

Basis

Basis, definition

Lemma.

The choice of definition and lemma

Context