Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called basic open sets, are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.
Not all families of subsets form a base for a topology. For example, because X is always an open subset of every topology on X, if a family B of subsets is to be a base for a topology on X then it must cover X, which by definition means that the union of all sets in B must be equal to X. If X has more than one point then there exist families of subsets of X that do not cover X and consequently, they can not form a basis for any topology on X. A family B of subsets of X that does form a basis for some topology on X is called a base for a topology on X, in which case this necessarily unique topology, call it τ, is said to be generated by B and B is consequently a basis for the topology τ. Such families of sets are frequently used to define topologies. A weaker notion related to bases is that of a subbasis for a topology. Bases for topologies are closely related to neighborhood bases.
A base is a collection B of open subsets of X satisfying the following properties:
An equivalent property is: any finite intersection[note 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.
The set Γ of all open intervals in ℝ form a basis for the Euclidean topology on ℝ. Every topology τ on a set X is a basis for itself (that is, τ is a basis for τ). Because of this, if a theorem's hypotheses assumes that a topology τ has some basis Γ, then this theorem can be applied using Γ := τ.
The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual basis of this topology, every finite intersection of basis elements is a basis element. Therefore bases are sometimes required to be stable by finite intersection.
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.
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 to be able to tell what sort of bases and local bases can exist. We have the 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+.
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.