Base (topology)


In mathematics, a base of a topology on a set is a collection of subsets of such that every finite intersection of elements of is a union of elements of. A base defines a topology on that has, as open sets, all unions of elements of.
Bases have been introduced because some topologies have a base consisting of open sets that have specific useful properties. This is typically the case for the Zariski topology on the spectrum of a ring. 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.

Definition and basic properties

A base is a collection B of subsets of X satisfying the following properties:
  1. The base elements cover X.
  2. Let B1, B2 be base elements and let I be their intersection. Then for each x in I, there is a base element B3 containing x such that B3 is subset of I.
An equivalent property is: any finite intersection 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. 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 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 and, 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, and would be in the topology generated by S, being unions of a single base element, and so their intersection would be as well. But 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.

Objects defined in terms of bases

s 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
  1. ∩F = ∅
  2. For each F1 and F2 in F the union F1 ∪ F2 is the intersection of some subfamily of F.
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.

Weight and character

We shall work with notions established in.
Fix X a topological space. Here, a network is a family of sets, for which, for all points x and open neighbourhoods U containing x, there exists B in for which x ∈ B ⊆ U. Note that, unlike a basis, the sets in a network need not be open.
We define the weight, w, as the minimum cardinality of a basis; we define the network weight, nw, as the minimum cardinality of a network; the character of a point,, as the minimum cardinality of a neighbourhood basis for x in X; and the character of X to be
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:
The last fact follows from f being compact Hausdorff, and hence ; as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable.

Increasing chains of open sets

Using the above notation, suppose that w ≤ κ some infinite cardinal. Then there does not exist a strictly increasing sequence of open sets of length ≥ κ+.
To see this, 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 = f = 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.