In mathematics, a base (or basis) B for a topological space X with topology T is a collection of sets in X such that every open set in X can be written as a union of elements of B.^[3][4][1] We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them.

A base is a collection B of subsets of X satisfying the following properties:

An equivalent property is: any finite intersection^[2] of elements of B can be written as a union of elements of B. These two conditions are exactly what is needed to ensure that the set of all unions of subsets of B is a topology on X.

If a collection B of subsets of X fails to satisfy these properties, then it is not a base for any topology on X. (It is a subbase, however, as is any collection of subsets of X.) Conversely, if B satisfies these properties, then there is a unique topology on X for which B is a base; it is called the topology generated by B. (This topology is the intersection of all topologies on X containing B.) This is a very common way of defining topologies. A sufficient but not necessary condition for B to generate a topology on X is that B is closed under intersections; then we can always take B3 = I above.

For example, the collection of all open intervals in the real line forms a base for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty. In fact they are a base for the standard topology on the real numbers.

However, a base is not unique. Many different bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals. In contrast to a basis of a vector space in linear algebra, a base need not be maximal; indeed, the only maximal base is the topology itself. In fact, any open set generated by a base may be safely added to the base without changing the topology. The smallest possible cardinality of a base is called the weight of the topological space.

An example of a collection of open sets which is not a base is the set S of all semi-infinite intervals of the forms (−∞, a) and (a, ∞), where a is a real number. Then S is not a base for any topology on R. To show this, suppose it were. Then, for example, (−∞, 1) and (0, ∞) would be in the topology generated by S, being unions of a single base element, and so their intersection (0,1) would be as well. But (0, 1) clearly cannot be written as a union of elements of S. Using the alternate definition, the second property fails, since no base element can "fit" inside this intersection.

Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

Closed sets are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space X, a family of closed sets F forms a base for the closed sets if and only if for each closed set A and each point x not in A there exists an element of F containing A but not containing x.

It is easy to check that F is a base for the closed sets of X if and only if the family of complements of members of F is a base for the open sets of X.

Let F be a base for the closed sets of X. Then

Any collection of subsets of a set X satisfying these properties forms a base for the closed sets of a topology on X. The closed sets of this topology are precisely the intersections of members of F.

In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space is completely regular if and only if the zero sets form a base for the closed sets. Given any topological space X, the zero sets form the base for the closed sets of some topology on X. This topology will be the finest completely regular topology on X coarser than the original one. In a similar vein, the Zariski topology on An is defined by taking the zero sets of polynomial functions as a base for the closed sets.

We shall work with notions established in (Engelking 1977, p. 12, pp. 127-128).

The point of computing the character and weight is useful to be able to tell what sort of bases and local bases can exist. We have following facts:

Using the above notation, suppose that w(X) ≤ κ some infinite cardinal. Then there does not exist a strictly increasing sequence of open sets (equivalently strictly decreasing sequence of closed sets) of length ≥ κ+.

To see this (without the axiom of choice), fix

as a basis of open sets. And suppose per contra, that

were a strictly increasing sequence of open sets. This means

For

we may use the basis to find some Uγ with x in Uγ ⊆ Vα. In this way we may well-define a map, f : κ+ → κ mapping each α to the least γ for which Uγ ⊆ Vα and meets

This map is injective, otherwise there would be α < β with f(α) = f(β) = γ, which would further imply Uγ ⊆ Vα but also meets

which is a contradiction. But this would go to show that κ+ ≤ κ, a contradiction.