Term: Subbase
Definition and Properties of Subbase
– A subbase of a topological space X is a subcollection B that generates the topology au of X.
– The subbase B is the smallest topology containing B.
– The collection of open sets consisting of all finite intersections of elements of B, together with X, forms a basis for au.
– For any subcollection S of the power set of X, there is a unique topology having S as a subbase.
– There is no unique subbasis for a given topology.
– A slightly different definition of subbase requires that the subbase B covers X.
– X is the union of all sets contained in B.
– This definition avoids confusion regarding the use of nullary intersections.
Examples and Special Cases of Subbases
– The topology generated by any subset S of {∅, X} is equal to the trivial topology {∅, X}.
– If au is a topology on X and B is a basis for au, then the topology generated by B is au.
– Any basis B for a topology au is also a subbasis for au.
– If S is any subset of au, then the topology generated by S will be a subset of au.
– The intervals (a, b), where a and b are rational, are a basis for the usual Euclidean topology.
– The product topology is a special case of the initial topology where the family of functions is the set of projections from the product to each factor.
– The subspace topology is a special case of the initial 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 a subbase consisting of functions V(K, U) where K is compact and U is an open subset of Y.
Alexander Subbase Theorem
– The Alexander Subbase Theorem states that if X has a subbasis S such that every cover of X by elements from S has a finite subcover, then X is compact.
– The converse to this theorem also holds.
– The Alexander Subbase Theorem is a significant result concerning subbases and compactness.
– The corresponding result for basic open covers is easier to prove.
– The proof of the Alexander Subbase Theorem uses Zorn’s Lemma and the maximality of a certain open cover.
Compactness of X and Use of Zorn’s Lemma
– The proof of compactness of X relies on the Ultrafilter principle.
– X is defined as a compact space.
– The proof uses the concept of covers and subcovers.
– The assumption that X is not compact leads to a contradiction.
– Therefore, X must be compact.
– The proof uses Zorn’s Lemma to establish the existence of a finite subcover.
– Zorn’s Lemma is a powerful tool in set theory.
– The proof only requires the intermediate strength of choice.
– The Ultrafilter principle is used instead of the full strength of choice.
– The proof shows that Zorn’s Lemma is sufficient to prove compactness.
Application to Tychonoff’s Theorem and Axiom of Choice
– The Alexander Subbase Theorem can be used to prove Tychonoff’s theorem.
– Bounded closed intervals in ℝ are compact.
– The subbase for ℝ can be used to prove this.
– Tychonoff’s theorem states that the product of compact spaces is compact.
– The proof of Tychonoff’s theorem becomes easier with the Alexander Subbase Theorem.
– The last step of the proof implicitly uses the axiom of choice.
– The existence of certain elements is ensured by the axiom of choice.
– The axiom of choice is equivalent to Zorn’s Lemma.
– The existence of elements is crucial in the proof.
– Rudin’s book provides further details on this topic.
References:
– Bourbaki’s ‘General Topology: Chapters 1–4’ is a comprehensive resource.
– Dugundji’s ‘Topology’ is a classic book on the subject.
– Munkres’ ‘Topology’ is a widely used textbook.
– Rudin’s ‘Functional Analysis’ covers advanced topics in topology.
– Willard’s ‘General Topology’ is a popular introduction to the subject.
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space with topology is a subcollection of that generates in the sense that is the smallest topology containing 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.
