In mathematics, a base (or basis) for the topology τ of a topological space (X, τ) is a family of open subsets of X such that every open set of the topology is equal to the union of some sub-family of . For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.

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.[1] 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 of a set form a base for a topology on . Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on , obtained by taking all possibly unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood bases.

Definition and basic properties

Given a topological space , a base[2][3][4][5] (or basis[6]) for the topology (also called a base for if the topology is understood) is a family of open sets such that every open set of the topology can be represented as the union of some subfamily of .[note 1] The elements of are called basic open sets. Equivalently, a family of subsets of is a base for the topology if and only if and for every open set in and point there is some basic open set such that .

For example, the collection of all open intervals in the real line forms a base for the standard topology on the real numbers. More generally, in a metric space the collection of all open balls about points of forms a base for the topology.

In general, a topological space can have many bases. The whole topology is always a base for itself (that is, is a base for ). For the real line, the collection of all open intervals is a base for the topology. So is the collection of all open intervals with rational endpoints, or the collection of all open intervals with irrational endpoints, for example. Note that two different bases need not have any basic open set in common. One of the topological properties of a space is the minimum cardinality of a base for its topology, called the weight of and denoted . From the examples above, the real line has countable weight.

If is a base for the topology of a space , it satisfies the following properties:[7][3]

(B1) The elements of cover , i.e., every point belongs to some element of .
(B2) For every and every point , there exists some such that .

Property (B1) corresponds to the fact that is an open set; property (B2) corresponds to the fact that is an open set.

Conversely, suppose is just a set without any topology and is a family of subsets of satisfying properties (B1) and (B2). Then is a base for the topology that it generates. More precisely, let be the family of all subsets of that are unions of subfamilies of . Then is a topology on and is a base for .[7][8] (Sketch: defines a topology because it is stable under arbitrary unions by construction, it is stable under finite intersections by (B2), it contains by (B1), and it contains the empty set as the union of the empty subfamily of . The family is then a base for by construction.) Such families of sets are a very common way of defining a topology.

In general, if is a set and is an arbitrary collection of subsets of , there is a (unique) smallest topology on containing . (This topology is the intersection of all topologies on containing .) The topology is called the topology generated by , and is called a subbase for . The topology can also be characterized as the set of all arbitrary unions of finite intersections of elements of . (See the article about subbase.) Now, if also satisfies properties (B1) and (B2), the topology generated by can be described in a simpler way without having to take intersections: is the set of all unions of elements of (and is base for in that case).

There is often an easy way to check condition (B2). If the intersection of any two elements of is itself an element of or is empty, then condition (B2) is automatically satisfied (by taking ). For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.

An example of a collection of open sets that is not a base is the set of all semi-infinite intervals of the forms and with . The topology generated by contains all open intervals , hence generates the standard topology on the real line. But is only a subbase for the topology, not a base: a finite open interval does not contain any element of (equivalently, property (B2) does not hold).

Examples

The set Γ of all open intervals in form a basis for the Euclidean topology on .

A non-empty family of subsets of a set X that is closed under finite intersections of two or more sets, which is called a π-system on X, is necessarily a base for a topology on X if and only if it covers X. By definition, every σ-algebra, every filter (and so in particular, every neighborhood filter), and every topology is a covering π-system and so also a base for a topology. In fact, if Γ is a filter on X then { ∅ } ∪ Γ is a topology on X and Γ is a basis for it. A base for a topology does not have to be closed under finite intersections and many aren't. But nevertheless, many topologies are defined by bases that are also closed under finite intersections. For example, each of the following families of subset of is closed under finite intersections and so each forms a basis for some< topology on :

Objects defined in terms of bases

The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set.

Theorems

Base for the closed sets

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 a family of closed sets forms a base for the closed sets if and only if for each closed set and each point not in there exists an element of containing but not containing A family is a base for the closed sets of if and only if its dual in that is the family of complements of members of , is a base for the open sets of

Let be a base for the closed sets of Then

  1. For each the union is the intersection of some subfamily of (that is, for any not in there is some containing and not containing ).

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

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 the zero sets form the base for the closed sets of some topology on This topology will be the finest completely regular topology on 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 (Engelking 1977, p. 12, pp. 127-128).

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 Note that, unlike a basis, the sets in a network need not be open.

We define the weight, w(X), as the minimum cardinality of a basis; we define the network weight, nw(X), 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(X) being compact Hausdorff, and hence (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable. (An application of this, for instance, is that every path in an Hausdorff space is compact metrisable.)

Increasing chains of open sets

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 UγVα. In this way we may well-define a map, f : κ+κ mapping each α to the least γ for which UγVα and meets

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

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

See also

Notes

  1. ^ The empty set, which is always open, is the union of the empty family.

References

  1. ^ Adams & Franzosa 2009, pp. 46–56.
  2. ^ Willard, Definition 5.1
  3. ^ a b Engelking, p. 12
  4. ^ Bourbaki, Definition 6, p. 21
  5. ^ Arkhangel'skii & Ponomarev, p. 40
  6. ^ Dugundji, Definition 2.1, p. 64
  7. ^ a b Willard, Theorem 5.3
  8. ^ Engelking, Proposition 1.2.1

Bibliography

  • Adams, Colin; Franzosa, Robert (2009). Introduction to Topology: Pure and Applied. New Delhi: Pearson Education. ISBN 978-81-317-2692-1. OCLC 789880519.
  • Arkhangel'skij, A.V.; Ponomarev, V.I. (1984). Fundamentals of general topology: problems and exercises. Mathematics and Its Applications. Vol. 13. Translated from the Russian by V. K. Jain. Dordrecht: D. Reidel Publishing. Zbl 0568.54001.
  • Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
  • Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
  • Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4
  • Willard, Stephen (2004) [1970]. General Topology (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.