Subbase
Encyclopedia
In topology
, a subbase (or subbasis) for a topological space
X with topology
T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.
(Note that if we use the nullary intersection convention, then there is no need to include X in the second definition.)
For any subcollection S of the power set P(X), there is a unique topology having S as a subbase. In particular, the intersection
of all topologies on X containing S satisfies this condition. In general, however, 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 P(X) and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.
However, with this definition, the two definitions above are not always equivalent. In other words, there exist spaces X with topology T, such that there exists a subcollection B of T such that T is the smallest topology containing B, yet B does not cover X. In practice, this a rare occurrence; e.g. a subbase of a space satisfying the T1 separation axiom must be a cover of that space.
s R has a subbase consisting of all semi-infinite
open intervals either of the form (−∞,a) or (b,∞), where a and b are real numbers. Together, these generate the usual topology, since the intersections
for a < b generate the usual topology. A second subbase is formed by taking the subfamily where a and b are rational
. The second subbase generates the usual topology as well, since the open intervals (a,b) with a, b rational, are a basis for the usual Euclidean topology.
The subbase consisting of all semi-infinite open intervals of the form (−∞,a) alone, where a is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom
, since all open sets have a non-empty intersection.
The initial 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 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 X to Y has for a subbase the set of functions
where K is compact
and U is open Y.
f−1(U) is open in X for each U in B.
.
Theorem: Let X be a topological space with a subbasis B. If every subbasic cover from B 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 an open 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. Q.E.D.
Although this proof makes use of Zorn's Lemma
, the proof does not need the full strength of choice. Instead, it relies of the intermediate Ultrafilter principle.
Using this theorem with the subbase for R above, one can give a very easy proof that bounded closed intervals in 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.
Note, that in the last step we implicitly used the axiom of choice (which is actually equivalent to Zorn's lemma
) to ensure the existence of ‹xi›.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a subbase (or subbasis) for a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X with topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.
Definition
Let X be a topological space with topology T. A subbase of T is usually defined as a subcollection B of T satisfying one of the two following equivalent conditions:- The subcollection B generates the topology T. This means that T is the smallest topology containing B: any topology U on X containing B must also contain T.
- The collection of open sets consisting of all finite intersectionsIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of elements of B, together with the set X and the empty set, forms a basis for T. This means that every non-empty proper open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in T can be written as a unionUnion (set theory)In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of finite intersections of elements of B. Explicitly, given a point x in a proper open set U, there are finitely many sets S1, …, Sn of B, such that the intersection of these sets contains x and is contained in U.
(Note that if we use the nullary intersection convention, then there is no need to include X in the second definition.)
For any subcollection S of the power set P(X), there is a unique topology having S as a subbase. In particular, the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of all topologies on X containing S satisfies this condition. In general, however, 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 P(X) and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.
Alternative definition
Sometimes, a slightly different definition of subbase is given which requires that the subbase B cover X. In this case, X is an open set in the topology generated, because it is the union of all the {Bi} as Bi ranges over B. This means that there can be no confusion regarding the use of nullary intersections in the definition.However, with this definition, the two definitions above are not always equivalent. In other words, there exist spaces X with topology T, such that there exists a subcollection B of T such that T is the smallest topology containing B, yet B does not cover X. In practice, this a rare occurrence; e.g. a subbase of a space satisfying the T1 separation axiom must be a cover of that space.
Examples
The usual topology on the real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s R has a subbase consisting of all semi-infinite
Semi-infinite
The term semi-infinite has several related meanings in various branches of pure and applied mathematics. It typically describes objects which are infinite or unbounded in some but not all possible ways.-In ordered structures and Euclidean spaces:...
open intervals either of the form (−∞,a) or (b,∞), where a and b are real numbers. Together, these generate the usual topology, since the intersections
for a < b generate the usual topology. A second subbase is formed by taking the subfamily where a and b are rationalRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
. The second subbase generates the usual topology as well, since the open intervals (a,b) with a, b rational, are a basis for the usual Euclidean topology.
The subbase consisting of all semi-infinite open intervals of the form (−∞,a) alone, where a is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
, since all open sets have a non-empty intersection.
The initial topology
Initial topology
In general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...
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 initial topology are the product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
, where the family of functions is the set of projections from the product to each factor, and the subspace topology
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
, where the family consists of just one function, the inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
.
The compact-open topology
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
on the space of continuous functions from X to Y has for a subbase the set of functions
where K is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
and U is open Y.
Results using subbases
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 iffIFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
f−1(U) is open in X for each U in B.
Alexander subbase theorem
There is one significant result concerning subbases, due to James Waddell Alexander IIJames Waddell Alexander II
James Waddell Alexander II was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others...
.
Theorem: Let X be a topological space with a subbasis B. If every subbasic cover from B has a finite subcover, then the space is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
.
(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
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
to find an open 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. Q.E.D.
Q.E.D.
Q.E.D. is an initialism of the Latin phrase , which translates as "which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out —...
Although this proof makes use of Zorn's Lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
, the proof does not need the full strength of choice. Instead, it relies of the intermediate Ultrafilter principle.
Using this theorem with the subbase for R above, one can give a very easy proof that bounded closed intervals in R are compact.
Tychonoff's theorem
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
, 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.
Note, that in the last step we implicitly used the axiom of choice (which is actually equivalent to Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
) to ensure the existence of ‹xi›.