Base (topology)

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In mathematics, a base (or basis; pl.: bases) 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 possible 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

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Given a topological space  , a base[2] (or basis[3]) 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:[4]

(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  .[5] (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

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The set Γ of all open intervals in   forms 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 are not. 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  :

  • The set Γ of all bounded open intervals in   generates the usual Euclidean topology on  .
  • The set Σ of all bounded closed intervals in   generates the discrete topology on   and so the Euclidean topology is a subset of this topology. This is despite the fact that Γ is not a subset of Σ. Consequently, the topology generated by Γ, which is the Euclidean topology on  , is coarser than the topology generated by Σ. In fact, it is strictly coarser because Σ contains non-empty compact sets which are never open in the Euclidean topology.
  • The set Γ  of all intervals in Γ such that both endpoints of the interval are rational numbers generates the same topology as Γ. This remains true if each instance of the symbol Γ is replaced by Σ.
  • Σ = { [r, ∞) : r  } generates a topology that is strictly coarser than the topology generated by Σ. No element of Σ is open in the Euclidean topology on  .
  • Γ = { (r, ∞) : r  } generates a topology that is strictly coarser than both the Euclidean topology and the topology generated by Σ. The sets Σ and Γ are disjoint, but nevertheless Γ is a subset of the topology generated by Σ.

Objects defined in terms of bases

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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

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  • A topology   is finer than a topology   if and only if for each   and each basic open set   of   containing  , there is a basic open set of   containing   and contained in  .
  • If   are bases for the topologies   then the collection of all set products   with each   is a base for the product topology   In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.
  • Let   be a base for   and let   be a subspace of  . Then if we intersect each element of   with  , the resulting collection of sets is a base for the subspace  .
  • If a function   maps every basic open set of   into an open set of  , it is an open map. Similarly, if every preimage of a basic open set of   is open in  , then   is continuous.
  •   is a base for a topological space   if and only if the subcollection of elements of   which contain   form a local base at  , for any point  .

Base for the closed sets

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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.  
  2. 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

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We shall work with notions established in (Engelking 1989, 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:

  • nw(X) ≤ w(X).
  • if X is discrete, then w(X) = nw(X) = |X|.
  • if X is Hausdorff, then nw(X) is finite if and only if X is finite discrete.
  • if B is a basis of X then there is a basis   of size  
  • if N a neighbourhood basis for x in X then there is a neighbourhood basis   of size  
  • if   is a continuous surjection, then nw(Y) ≤ w(X). (Simply consider the Y-network   for each basis B of X.)
  • if   is Hausdorff, then there exists a weaker Hausdorff topology   so that   So a fortiori, if X is also compact, then such topologies coincide and hence we have, combined with the first fact, nw(X) = w(X).
  • if   a continuous surjective map from a compact metrizable space to an Hausdorff space, then Y is compact metrizable.

The last fact follows from f(X) being compact Hausdorff, and hence   (since compact metrizable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrizable exactly in case they are second countable. (An application of this, for instance, is that every path in a Hausdorff space is compact metrizable.)

Increasing chains of open sets

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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

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Notes

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

References

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  1. ^ Adams & Franzosa 2009, pp. 46–56.
  2. ^ Willard 2004, Definition 5.1; Engelking 1989, p. 12; Bourbaki 1989, Definition 6, p. 21; Arkhangel'skii & Ponomarev 1984, p. 40.
  3. ^ Dugundji 1966, Definition 2.1, p. 64.
  4. ^ Willard 2004, Theorem 5.3; Engelking 1989, p. 12.
  5. ^ Willard 2004, Theorem 5.3; Engelking 1989, Proposition 1.2.1.

Bibliography

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  • Adams, Colin; Franzosa, Robert (2009). Introduction to Topology: Pure and Applied. New Delhi: Pearson Education. ISBN 978-81-317-2692-1. OCLC 789880519.
  • Arkhangel'skii, 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 (1989). General topology. Berlin: Heldermann Verlag. ISBN 3-88538-006-4.
  • Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.