Editing Subbase (section)

==Definition==

Let <math>X</math> be a topological space with topology <math>\tau.</math> A '''subbase''' of <math>\tau</math> is usually defined as a subcollection <math>B</math> of <math>\tau</math> satisfying one of the three following equivalent conditions:

#<math>\tau</math> is the smallest topology containing <math>B</math>: any topology <math>\tau^\prime</math> on <math>X</math> containing <math>B</math> must also contain <math>\tau.</math>
#<math>\tau</math> is the [[Intersection (set theory)|intersection]] of all topologies on X containing <math>B.</math>
#The collection of open sets consisting of <math>X</math> and all finite [[Intersection (set theory)|intersections]] of elements of <math>B</math> forms a [[Basis (topology)|basis]] for <math>\tau.</math>{{sfn|Rudin|1991|p=392 Appendix A2}}<ref group=note>Rudin's definition is less general than ours, because it effectively requires that <math>B</math> covers <math>X</math> (see "Alternative definition" subsection below). We drop this requirement here, and assume that <math>B</math> is any subset of <math>\mathcal{P}(X)</math></ref> This means that every proper [[open set]] in <math>\tau</math> can be written as a [[Union (set theory)|union]] of finite intersections of elements of <math>B.</math> Explicitly, given a point <math>x</math> in an open set <math>U \subsetneq X,</math> there are finitely many sets <math>S_1, \ldots, S_n</math> of <math>B,</math> such that the intersection of these sets contains <math>x</math> and is contained in <math>U.</math>

If we additionally assume that <math>B</math> [[Cover (topology)|covers]] <math>X</math>, or if we use the [[nullary intersection]] convention, then there is no need to include <math>X</math> in the third definition.

If <math>B</math> is a subbase of <math>\tau</math>, we say that <math>B</math> '''generates''' the topology <math>\tau.</math> This terminology originates from the explicit construction of <math>\tau</math> from <math>B</math> using the second or third definition above.

Elements of subbase are called {{em|subbasic (open) sets}}. A [[Cover (topology)|cover]] composed of subbasic sets is called a {{em|subbasic (open) cover}}.

For {{em|any}} subcollection <math>S</math> of the [[power set]] <math>\wp(X),</math> there is a unique topology having <math>S</math> as a subbase; it is the intersection of all topologies on <math>X</math> containing <math>S</math>. In general, however, the converse is not true, i.e. there is no unique subbasis for a given topology.

Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set <math>\wp(X)</math> 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===

Less commonly, a slightly different definition of subbase is given which requires that the subbase <math>\mathcal{B}</math> cover <math>X.</math>{{sfn | Munkres | 2000 | pp=82}}  In this case, <math>X</math> is the union of all sets contained in <math>\mathcal{B}.</math> This means that there can be no confusion regarding the use of nullary intersections in the definition.

However, this definition is not always equivalent to the three definitions above. There exist topological spaces <math>(X, \tau)</math> with subcollections <math>\mathcal{B} \subseteq \tau</math> of the topology such that <math>\tau</math> is the smallest topology containing <math>\mathcal{B}</math>, yet <math>\mathcal{B}</math> does not cover <math>X</math>. For example, consider a topological space <math>(X,\tau)</math> with <math>\tau=\{\varnothing, \{p\}, X\}</math> and <math>\mathcal{B}=\{\{p\}\}</math> for some <math>p\in X.</math> Clearly, <math>\mathcal{B}</math> is a subbase of <math>\tau</math>, yet <math>\mathcal{B}</math> doesn't cover <math>X</math> as long as <math>X</math> has at least <math>2</math> 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|T<sub>1</sub> separation axiom]] must be a cover of that space.