This page discusses the classical case of bases for topological spaces. For bases on sites, that is for Grothendieck topologies, see at Grothendieck pretopology.
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A base or subbase for a topological space is a way of generating its topology from something simpler. This is the appliction to topology of the general concept of base.
Let $X$ be a topological space, and let $\tau$ be its collection of open subsets (its ‘topology’).
A base or basis for (or “of”) $X$ (or $\tau$) is a collection $B \subset \tau$ – whose members are called basic open subsets or generating open subsets – such that every open subset is a union of basic ones.
A subbase for (or “of”) $X$ (or $\tau$) is a subcollection $S \subset \tau$ – whose members are called subbasic open subsets – such that every open subset is a union of finitary intersections of subbasic ones.
If only the underlying set of $X$ is given, then a base or subbase on this set is any collection of subsets of $X$ that is a base or subbase for some topology on $X$. See below for a characterisation of which collections these can be.
Now fix a point $a$ in $X$.
A local base or base of neighborhoods or fundamental system of neighborhoods for (or “of”) $X$ (or $\tau$) at $a$ is a subcollection $B \subset \tau$ – whose members are called basic neighborhoods or generating neighborhoods of $a$ – such that every basic neighborhood of $a$ is a neighborhood and every neighborhood of $a$ 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 $a$ is a family of neighbourhoods $a$ such that each neighbourhood of $a$ contains a finite intersection of elements of the family.
The minimum cardinality of a base of $X$ is the weight of $X$. The minimum cardinality of a base of neighborhoods at $a$ is the character of $X$ at $a$. The supremum of the characters at all points of $X$ is the character of $X$.
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.
If the character of $X$ is countable, we say that $X$ satisfies the first axiom of countability; if the weight is countable, we say that $X$ satisfies the second axiom of countability.
For the discrete topology on a set $X$, the collection of all singleton subsets is a base, and the singleton $\{x\}$ is a local base at $x$. 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 $(a,b) \subset \mathbb{R}$.) Similarly, the collection of open balls containing a given point is a local basis at that point.
This means that covering families consisting of such basic open subsets are good open covers.
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 $(a,b) \subset \mathbb{R}$ where $b - a$ 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 $D$ which is countable. Now the space has a basis consisting of the open balls with rational radius and centres in $D$. (For instance, a base for the topology on the real line is given by the collection of open intervals $(a,b) \subset \mathbb{R}$ where $a$ and $b$ are rational.) Therefore, every separable metric space is second-countable.
Let $X$ be simply a set.
A collection $B$ of subsets of $X$ is a base for some topology on $X$ iff these conditions are met:
This is a sort of colax closure (see discussion) under finitary intersections.
Every collection $S$ of subsets of $X$ is a subbase for some topology on $X$.
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 $X$ as being encoded in the standard Grothendieck topology that it induces on its category of open subsets $Op(X)$, then a base for the topology induces a coverage on $Op(X)$, 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.