satisfying one of the two following equivalent conditions: (If we use the nullary intersection convention, then there is no need to include
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
We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.
This means that there can be no confusion regarding the use of nullary intersections in the definition.
But as seen below, to prove the Alexander subbase theorem,[3] one must assume that
[clarification needed] The topology generated by any subset
has a subbase consisting of all semi-infinite open intervals either of the form
A second subbase is formed by taking the subfamily where
The second subbase generates the usual topology as well, since the open intervals
rational, are a basis for the usual Euclidean topology.
The subbase consisting of all semi-infinite open intervals of the form
is a real number, does not generate the usual topology.
The resulting topology does not satisfy the T1 separation axiom, since if
Because continuity can be defined in terms of the inverse images of open sets, this means that the initial topology on
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
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.
The converse to this theorem also holds and it is proven by using
is an infinite set), yet every subbasic cover from
denote the set of all open covers of
by subset inclusion and use Zorn's Lemma to find an element
there must exist a finite collection of subbasic open sets
Instead, it relies on the intermediate Ultrafilter principle.
above, one can give a very easy proof that bounded closed intervals in
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.
has, by definition, a subbase consisting of cylinder sets that are the inverse projections of an open set in one factor.
into subfamilies that consist of exactly those cylinder sets corresponding to a given factor space.
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