Other Definitions
subbase (dict)
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SubbaseIn topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T such that the collection of finite intersections of elements of B form a basis for T. This means that every open set in T can be written as a union of finite intersections of elements of B. Equivalently, this means that given x∈U with U open, there are finitely many members S1, … ,Sn∈B such that x∈S1∩…∩Sn⊆U. We say that the subbase generates the topology T, and that T is generated by B. (Some authors disallow B to be empty. Others allow an empty B, using the nullary intersection convention to generate the indiscrete topology. The latter convention is followed here, since the resulting statements are simpler.) There is, in general, no unique subbasis for a given topology. For any subcollection S of the power set P(X), there is a unique smallest topology T containing S (namely, the arbitrary unions of the finite intersections of members of S). S forms a subbasis for this topology. It can also be defined from above as the intersection of all topologies on X containing S.) One nice fact about subbases is that continuity of a function need only be checked on a subbase of the range. That is, if B is a subbase for Y, a function f : X → Y is continuous iff f−1(U) is open in X for each U in B. Examples The usual topology on the real numbers R has a subbase consisting of all semi-infinite open intervals either of the form (−∞,a) or (a,∞), where a is a real number. A second subbase is formed by taking the subfamily where a is rational. The weak topology defined by a family of functions fi : X → Yi, where each Yi has a topology, is the coarsest topology on X such that each fi is continuous. Because continuity can be defined by the inverse images of open sets, this means that the weak topology on X is given by taking all fi−1(Ui), where Ui ranges over all open subsets of Yi, as a subbasis. Two important special cases of the weak 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 on the space of continuous functions from X to Y has for a subbase the set of functions -
where K is compact and U is open Y. Alexander subbase theorem There is one significant result concerning subbases, due to J. W. Alexander. Theorem: If every subbasic cover has a finite subcover, then the space is compact. (The corresponding result for basic covers is trivial.) Proof (outline): Assume by way of contradiction that the space X is not compact, yet every subbasic cover from B has a finite subcover. Use Zorn's Lemma to find a cover C without finite subcover that is maximal amongst such covers. That means that if V is not in C, then C∪{V} has a finite subcover, necessarily of the form C0∪{V}. Consider C∩B, that is, the subbasic subfamily of C. If it covered X, then by hypothesis, it would have a finite subcover. But C does not have such, so C∩B does not cover X. Let x∈X that is not covered. C covers X, so for U∈C, x∈U. B is a subbasis, so for some S1, … ,Sn∈B, x∈S1∩…∩Sn⊆U. Since x is uncovered, Si∉C. As noted above, this means that for each i, Si along with a finite subfamily Ci of C, covers X. But then U and all the Ci’s cover X, so C has a finite subcover after all. QED Using this theorem with the subbase for R above, one can give a very easy proof that finite closed intervals on R are compact. Tychonoff's theorem, that the product of compact spaces is compact, also has a short proof. The product topology on ∏iXi has, by definition, a subbase consisting of cylinder sets that are the inverse projections of an open set in one factor. Given a subbasic family C of the product that does not have a finite subcover, we can partition C=∪iCi into subfamilies that consist of exactly those cylinder sets corresponding to a given factor space. By assumption, no Ci has a finite subcover. Being cylinder sets, this means their projections onto Xi have no finite subcover, and since each Xi is compact, we can find a point xi∈Xi that is not covered by the projections of Ci onto Xi. But then ‹xi› is not covered by C. See also
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