Basis for a topological space
This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology
Definition when the topological space is specified
Definition with symbols
If is a topological space, a basis for is a collection of open subsets of (here, is an indexing set) such that for any open subset of , there exists such that:
Definition 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.
Definition with symbols
In symbols: if is a set, a collection of subsets of is said to form a basis for a topology on if the following two conditions are satisfied:
- For all , there exists such that:
The second condition is sometimes stated as follows: if , then there exists such that .
The topology generated by the s is defined as follows: a subset is open in if and only if there exists
Equivalence of definitions
Further information: Equivalence of definitions of basis
- For any topological space, the collection of all open subsets is a basis. That's because any open subset of a topological space can be expressed as a union of size one.
- For a discrete topological space, the collection of one-point subsets forms a basis. That's because every open subset of a discrete topological space is a union of one-point subsets, namely, the one-point subsets corresponding to its elements.
Examples from metric spaces
- Euclidean space: A basis for the usual topology on Euclidean space is the open balls. An open ball of radius centered at a point , is defined as the set of all whose distance from is strictly smaller than . By the way the topology on is defined, these open balls clearly form a basis.
- Metric space: Given any metric space, there is a natural way of viewing it as a topological space. This natural way involves declaring the collection of open balls in the metric space as a basis. To see that this gives a well-defined topology, we need to check that the collection of open balls satisfies the conditions to be a basis for a topological space. This follows from the conditions for a metric space, including nonnegativity, symmetry and the triangle inequality.