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
Two consequence of this definition are:
- Every element of
[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
- For each
, there is [...]. - If
where and are basis elements, then [...].