# 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.