In topology in mathematics, a subbase (or subbasis, prebase, prebasis) for the topology of a topological space as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

Subbase is a weaker notion than that of a base for a topology.

Definition

A subbase satisfying one of the three following equivalent conditions:

1. is the intersection

2. and all finite intersections forms a basis This means that every proper open set can be written as a union

covers, or if we use the nullary intersection in the third definition.

generates using the second or third definition above.

Elements of subbase are called . A cover composed of subbasic sets is called a .

For of the power set. In general, however, the converse is not true, i.e. there is no unique subbasis for a given topology.

and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the three conditions is more useful than the others.

Alternative definition

This means that there can be no confusion regarding the use of nullary intersections in the definition.

elements. In practice, this is a rare occurrence. E.g. a subbase of a space that has at least two points and satisfies the T1 separation axiom must be a cover of that space.

Examples

The usual topology on the real numbers has a subbase consisting of all semi-infinite are rational rational, are a basis for the usual Euclidean topology.

is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom every open set

The initial topology is continuous. Because continuity can be defined in terms of the inverse images as a subbasis.

Two important special cases of the initial topology are the product topology, where the family of functions is the set of projections from the product to each factor, and the subspace topology, where the family consists of just one function, the inclusion map.

The compact-open topology is compact

is a Hausdorff with the Euclidean topology be any non-empty denote the subspace topology (see the footnote for an explanation), is a proper subset is the smallest topology ).

Results using subbases

One nice fact about subbases is that continuity is continuous if and only if A net (or sequence if and only if every

Alexander subbase theorem

The Alexander Subbase Theorem is a significant result concerning subbases that is due to James Waddell Alexander II. The corresponding result for basic (rather than subbasic) open covers is much easier to prove.

1. Alexander subbase theorem: is compact.

)

1. has a finite subcover.

Although this proof makes use of Zorn's Lemma, the proof does not need the full strength of choice. Instead, it relies on the intermediate Ultrafilter principle.

are compact. More generally, Tychonoff's theorem, which states that the product of non-empty compact spaces is compact, has a short proof if the Alexander Subbase Theorem is used.

See also

Notes

Citations

References