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Subbase
Collection of subsets that generate a topology From Wikipedia, the free encyclopedia
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In topology, a subbase (or subbasis, prebase, prebasis) for the topology τ of a topological space (X, τ) 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.
Subbase is a weaker notion than that of a base for a topology.
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Definition
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Let be a topological space with topology A subbase of is usually defined as a subcollection of satisfying one of the three following equivalent conditions:
- is the smallest topology containing : any topology on containing must also contain
- is the intersection of all topologies on X containing
- The collection of open sets consisting of and all finite intersections of elements of forms a basis for [1][note 1] This means that every proper open set in can be written as a union of finite intersections of elements of Explicitly, given a point in an open set there are finitely many sets of such that the intersection of these sets contains and is contained in
If we additionally assume that covers , or if we use the nullary intersection convention, then there is no need to include in the third definition.
If is a subbase of , we say that generates the topology This terminology originates from the explicit construction of from using the second or third definition above.
Elements of subbase are called subbasic (open) sets. A cover composed of subbasic sets is called a subbasic (open) cover.
For any subcollection of the power set there is a unique topology having as a subbase; it is the intersection of all topologies on containing . 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 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 cover [2] In this case, is the union of all sets contained in 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 with subcollections of the topology such that is the smallest topology containing , yet does not cover . For example, consider a topological space with and for some Clearly, is a subbase of , yet doesn't cover as long as has at least 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.
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Examples
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The topology generated by any subset (including by the empty set ) is equal to the trivial topology
If is a topology on and is a basis for then the topology generated by is Thus any basis for a topology is also a subbasis for If is any subset of then the topology generated by will be a subset of
The usual topology on the real numbers has a subbase consisting of all semi-infinite open intervals either of the form or where and are real numbers. Together, these generate the usual topology, since the intersections for generate the usual topology. A second subbase is formed by taking the subfamily where and are rational. The second subbase generates the usual topology as well, since the open intervals with rational, are a basis for the usual Euclidean topology.
The subbase consisting of all semi-infinite open intervals of the form alone, where is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom, since if every open set containing also contains
The initial topology on defined by a family of functions where each has a topology, is the coarsest topology on such that each is continuous. Because continuity can be defined in terms of the inverse images of open sets, this means that the initial topology on is given by taking all where ranges over all open subsets of 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 on the space of continuous functions from to has for a subbase the set of functions where is compact and is an open subset of
Suppose that is a Hausdorff topological space with containing two or more elements (for example, with the Euclidean topology). Let be any non-empty open subset of (for example, could be a non-empty bounded open interval in ) and let denote the subspace topology on that inherits from (so ). Then the topology generated by on is equal to the union (see the footnote for an explanation), [note 2] where (since is Hausdorff, equality will hold if and only if ). Note that if is a proper subset of then is the smallest topology on containing yet does not cover (that is, the union is a proper subset of ).
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Results using subbases
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One nice fact about subbases is that continuity of a function need only be checked on a subbase of the range. That is, if is a map between topological spaces and if is a subbase for then is continuous if and only if is open in for every A net (or sequence) converges to a point if and only if every subbasic neighborhood of contains all for sufficiently large
Alexander subbase theorem
The Alexander Subbase Theorem is a significant result concerning subbases that is due to James Waddell Alexander II.[3] The corresponding result for basic (rather than subbasic) open covers is much easier to prove.
- Alexander subbase theorem:[3][1] Let be a topological space, and be a subbase of If every cover of by elements from has a finite subcover, then is compact.
The converse to this theorem also holds (because every cover of by elements of is an open cover of )
- Let be a topological space, and be a subbase of If is compact, then every cover of by elements from 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.[3]
Using this theorem with the subbase for above, one can give a very easy proof that bounded closed intervals in 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.
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See also
- Base (topology) – Collection of open sets used to define a topology
Notes
- Rudin's definition is less general than ours, because it effectively requires that covers (see "Alternative definition" subsection below). We drop this requirement here, and assume that is any subset of
- Since is a topology on and is an open subset of , it is easy to verify that is a topology on . In particular, is closed under unions and finite intersections because is. But since , is not a topology on an is clearly the smallest topology on containing ).
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Citations
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