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:

$V=\bigcup _{i\in J}U_{j}$

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:

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

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

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