In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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.

Contents 1 Simple properties of bases 2 Objects defined in terms of bases 3 Theorems 4 Base for the closed sets 5 See also 6 References

Simple properties of bases

Two important properties of bases are:

1. The base elements cover X. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection
   
2. Let B₁, B₂ be base elements and let I be their intersection. Then for each x in I, there is another base element B₃ containing x and contained in I.

If a collection B of subsets of X fails to satisfy either of these, then it is not a base for any topology on X. (It is a subbase, however, as is any collection of subsets of X. In Highway engineering, subbase is a layer between Subgrade and the Base course. ) Conversely, if B satisfies both of the conditions 1 and 2, 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. In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently ) 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 open intervals in the real line form 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 Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In fact they are a base for the standard topology on the real numbers. In Mathematics, the real numbers may be described informally in several different ways

However, a base is not unique. Many bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the real numbers, 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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Linear algebra is the branch of Mathematics concerned with In fact, any open sets in the space 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. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set"

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 the 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

• The order topology is usually defined as the topology generated by a collection of open-interval-like sets. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.
• The metric topology is usually defined as the topology generated by a collection of open balls. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric
• A second-countable space is one that has a countable base. In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability "
• The discrete topology has the singletons as a base. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " In Mathematics, a singleton is a set with exactly one element

Theorems

• For each point x in an open set U, there is a base element containing x and contained in U.
• A topology T₂ is finer than a topology T₁ if and only if for each x and each base element B of T₁ containing x, there is a base element of T₂ containing x and contained in B. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. ↔
• If B₁,B₂,. . . ,B_n are bases for the topologies T₁,T₂,. . . ,T_n, then the set product B₁ × B₂ × . Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. . . × B_n is a base for the product topology T₁ × T₂ × . In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural . . × T_n. 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 B be a base for X and let Y be a subspace of X. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Then if we intersect each element of B with Y, the resulting collection of sets is a base for the subspace Y.
• If a function f:X → Y maps every base element of X into an open set of Y, it is an open map. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets Similarly, if every preimage of a base element of Y is open in X, then f is continuous. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function
• A collection of subsets of X is a topology on X if and only if it generates itself.
• B is a basis for a topological space X if and only if its elements can be used to form a local base for any point x of X. In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the

Base for the closed sets

Closed sets are equally adept at describing the topology of a space. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F. In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently

Equivalently, a family of closed sets forms a base for the closed sets 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. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation

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 (i. e. for any x not in F₁ or F₂ there is an F₃ in F containing F₁ ∪ F₂ and not containing 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 if completely regular if and only if the zero sets form a base for the closed sets. In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces Zero Set is the one and only album by German electronic music trio of Dieter Moebius, Conny Plank, and Mani Neumeier. Given any topological space X, the zero sets form the base for the closed sets of some topology on X. This topology will be 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. In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic

See also

• Subbase
• Gluing axiom

References

• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts. In Highway engineering, subbase is a layer between Subgrade and the Base course. In Mathematics, the gluing axiom is introduced to define what a sheaf F on a Topological space X must satisfy given that it is a
• Munkres, Topology: a First Course, Prentice-Hall, 1975.

---

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic