If only the underlying set of is given, then a base or subbase on this set is any collection of subsets of that is a base or subbase for some topology on . See below for a characterisation of which collections these can be.
Now fix a point in .
A local base or base of neighborhoods or fundamental system of neighborhoods for (or “of”) (or ) at is a subcollection – whose members are called basic neighborhoods or generating neighborhoods of – such that every basic neighborhood of is a neighborhood and every neighborhood of is a superset of some basic neighborhood.
We may also allow basic neighborhoods to be non-open, but this really doesn't make any difference; any local base may be refined to a local base of open neighborhoods, and most local bases in practices already come that way.
A local subbase at is a family of neighbourhoods such that each neighbourhood of contains a finite intersection of elements of the family.
The minimum cardinality of a base of is the weight of . The minimum cardinality of a base of neighborhoods at is the character of at . The supremum of the characters at all points of is the character of .
We have assumed the axiom of choice to simplify the description of this concept; but in general one must speak of classes of cardinalities rather than individual cardinalities.
For the discrete topology on a set , the collection of all singleton subsets is a base, and the singleton is a local base at . Thus every discrete space is first-countable, but only countable discrete spaces are second-countable.
For every metric space, in particular every paracompact Riemannian manifold, the collection of open subsets that are open balls forms a base for the topology. (For instance, a base for the topology on the real line is given by the collection of open intervals .) Similarly, the collection of open balls containing a given point is a local basis at that point.
Refining the previous example, every metric space has a basis consisting of the open balls with rational radius. (For instance, a base for the topology on the real line is given by the collection of open intervals where is rational.) Similarly, the collection of open balls with rational radius containing a given point is a local base at that point. Therefore, every metric space is first-countable.
Now consider a separable metric space; that is, we have a dense subset which is countable. Now the space has a basis consisting of the open balls with rational radius and centres in . (For instance, a base for the topology on the real line is given by the collection of open intervals where and are rational.) Therefore, every separable metric space is second-countable.
Let be simply a set.
A collection of subsets of is a base for some topology on iff these conditions are met:
Every collection of subsets of is a subbase for some topology on .
A subbase naturally generates a base (for the same topology) by closing it under finitary intersections. (The resulting base will actually be closed under intersection, not just colax-closed.)
If one thinks of the topology on as being encoded in the standard Grothendieck topology that it induces on its category of open subsets , then a base for the topology induces a coverage on , whose covering families are the open covers by basic open subsets, which generates this Grothendieck topology.
This coverage is not in general a basis for the Grothendieck topology, because a base for a topological space is in general not closed under intersection with arbitrary open subsets; a coverage is only a basis if is stable under pullback (here, closed under these intersections) and transitive. Unfortunately the established terminology “basis” in topology and topos theory is not quite consistent with the inclusion of topological spaces into topos theory: “basis” in topology corresponds to “coverage” in topos theory, not to “basis” in topos theory.