Basis for a topological space
This article is about a basic definition in topology.
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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
The topology generated by this basis is the topology in which the open sets are precisely the unions of basis sets.
(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).