Basis for a topological space
Definition
When the topological space is pre-specified
A basis for a topological space is a collection of open subsets of the topological space, such that every open subset can be expressed as a (possibly empty) union of basis subsets.
When the topological space is not specified
Given a set, a collection of subsets of the set is said to form a basis for a topological space if the following two conditions are satisfied:
- The union of all members of the collection is the whole space
- Any finite intersection of members of the collection, is itself a union of members of the collection
(Any basis for a topological space as per the first definition, must satisfy the above two conditions, by the axioms that a topology on a set must satisfy).