Collection of subsets that generate a topology
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.
Definition
Let
be a topological space with topology
A subbase of
is usually defined as a subcollection
of
satisfying one of the two following equivalent conditions:
- The subcollection
generates the topology
This means that
is the smallest topology containing
: any topology
on
containing
must also contain ![{\displaystyle \tau .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- The collection of open sets consisting of all finite intersections of elements of
forms a basis for
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 ![{\displaystyle U.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
(If we use the nullary intersection convention, then there is no need to include
in the second definition.)
For any subcollection
of the power set
there is a unique topology having
as a subbase. In particular, the intersection of all topologies on
containing
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
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
Less commonly, a slightly different definition of subbase is given which requires that the subbase
cover
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 two definitions above. There exist topological spaces
with subcollections
of the topology such that
is the smallest topology containing
, yet
does not cover
. (An example is given at the end of the next section.) 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. But as seen below, to prove the Alexander subbase theorem,[3] one must assume that
covers
[clarification needed]
Examples
The topology generated by any subset
(including by the empty set
) is equal to the trivial topology ![{\displaystyle \{\varnothing ,X\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 ![{\displaystyle \tau .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 ![{\displaystyle a.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 ![{\displaystyle Y.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 1]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
).
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
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 ![{\displaystyle i\in I.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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] Let
be a topological space. If
has a subbasis
such that every cover of
by elements from
has a finite subcover, then
is compact.
The converse to this theorem also holds and it is proven by using
(since every topology is a subbasis for itself).
- If
is compact and
is a subbasis for
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.
See also
Notes
- ^ 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
).
Citations
- ^ a b c d Muger, Michael (2020). Topology for the Working Mathematician.
References