# Dictionary Definition

subbase n : the lowest molding of an
architectural base or of a baseboard

# Extensive Definition

- In highway engineering, subbase is a layer between subgrade and the base course.

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.

## 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 made up with X and all finite intersections of elements of B forms a basis for T. This means that every non-empty proper open set in T can be written as a union 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
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 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 axiom must
be a cover of that space.

## Examples

The usual topology on the real numbers
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 (a,b) = (-\infty,b) \cap (a,\infty) 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

- V(K,U) = \

## 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
iff f−1(U) is open in X for
each U in B.

### Alexander subbase theorem

There is one significant result concerning
subbases, due to J. W.
Alexander.

Theorem: If every subbasic cover 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∪ has
a finite subcover, necessarily of the form C0∪.

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.

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.

## References

- Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.

subbase in Italian: Prebase

subbase in Polish: Podbaza