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basis (topology) (Definition)

Let $ (X,\mathcal{T})$ be a topological space. A subset $ \mathcal{B}$ of $ \mathcal{T}$ is a basis for $ \mathcal{T}$ if every member of $ \mathcal{T}$ is a union of members of $ \mathcal{B}$.

Equivalently, $ \mathcal{B}$ is a basis if and only if whenever $ U$ is open and $ x\in U$ then there is an open set $ V \in \mathcal{B}$ such that $ x\in V\subseteq U$.

The topology generated by a basis $ \mathcal{B}$ consists of exactly the unions of the elements of $ \mathcal{B}$.

We also have the following easy characterization: (for a proof, see the attachment)

Proposition 1   A collection of subsets $ \mathcal{B}$ of $ X$ is a basis for some topology on $ X$ if and only if each $ x \in X$ is in some element $ B \in \mathcal{B}$ and whenever $ B_1, B_2\in\mathcal{B}$ and $ x\in B_1\cap B_2$ then there is $ B_3\in\mathcal{B}$ such that $ x\in B_3\subseteq B_1\cap B_2$.

Examples

1. A basis for the usual topology of the real line is given by the set of open intervals since every open set can be expressed as a union of open intervals. One may choose a smaller set as a basis. For instance, the set of all open intervals with rational endpoints and the set of all intervals whose length is a power of $ 1/2$ are also bases. However, the set of all open intervals of length $ 1$ is not a basis although it is a subbasis (since any interval of length less than $ 1$ can be expressed as an intersection of two intervals of length $ 1$).

2. More generally, the set of open balls forms a basis for the topology on a metric space.

3. The set of all subsets with one element forms a basis for the discrete topology on any set.



"basis (topology)" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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See Also: subbasis, a compact metric space is second countable

Other names:  basis, base, topology generated by a basis
Keywords:  topology

Attachments:
conditions for a collection of subsets to be a basis for some topology (Proof) by waj
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Cross-references: discrete topology, metric space, open balls, intersection, subbasis, bases, power, length, intervals, endpoints, rational, open intervals, line, real, usual topology, collection, characterization, open, union, subset, topological space
There are 119 references to this entry.

This is version 12 of basis (topology), born on 2002-01-01, modified 2007-12-08.
Object id is 1161, canonical name is BasisTopologicalSpace.
Accessed 13519 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
Pending Errata and Addenda
None.
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Discussion
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about proposition 1 by nime on 2004-04-04 21:50:36
there is at least one condition missing, since even the empty set fullfills the conditions of proposition 1. But a basis must have one element, (if X ist not empty).
[ reply | up ]
V ? by awmorp on 2004-03-29 22:19:11
Line 2: 'Equivalently, B is a basis if and only if whenever U is open and x is an element of U then there is an open set V such that x is an element of V is a subset of U.'

What is the relationship between V and B? Should it be 'there is an open set V which is an element of B such that..' ?
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