Basis for a topological space

This article is about a basic definition in topology.
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Definition when the topological space is specified

Symbol-free definition

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.

Definition with symbols

If $X$ is a topological space, a basis for $X$ is a collection ${U_{i}}_{i \in I}$ of open subsets of $X$ (here, $I$ is an indexing set) such that for any open subset $V$ of $X$ , there exists $J \subset I$ such that:

$V = ⋃_{i \in J} U_{j}$

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 or a basis for a topology 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.

In symbols: if $X$ is a set, a collection ${U_{i}}_{i \in I}$ of subsets of $X$ is said to form a basis for a topology on $X$ if the following two conditions are satisfied:

$⋃_{i \in I} U_{i} = X$
For all $i, j \in I$ , there exists $J \subset I$ such that:

$U_{i} \cap U_{j} = ⋂_{k \in J} U_{k}$

The topology generated by the $U_{i}$ s is defined as follows: a subset $V$ is open in $X$ if and only if there exists

Equivalence of definitions

Further information: Equivalence of definitions of basis

Related notions

Subbasis for a topological space