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. 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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

Simple properties of bases

Two important properties of bases which together form an alternate definition are:

• The base elements cover X.
• 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.) Conversely, if B satisfies each of these conditions, then there is a unique topology on X for which B is a base. (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. In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X and... In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations... 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, all elements of B that also belong to A), but no other elements. ...

If we are given a topological space, we can verify whether or not some collection of open sets is a base for the space either using the above or directly from the definition. For example, given the standard topology on the real numbers, we know the open intervals are open. In fact, they are a base, because the intersection of any two open intervals is itself an open interval or empty. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

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. 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 weight of the topological space. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...

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.

Objects defined in terms of bases

• The order topology is usually defined as the topology generated by a collection of open-interval-like sets.
• The metric topology is usually defined as the topology generated by a collection of open balls.
• A second-countable space is one that has a countable base.
• The discrete topology has the singletons as a basis.

In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ... In mathematics the term countable set is used to describe the size of a set, e. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... 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.
• If B₁,B₂,...,B_n are bases for the topologies T₁,T₂,...,T_n, then the set product B₁ × B₂ × ... × B_n is a base for the topology T₁ × T₂ × ... × 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 is a base for X and let Y be a subspace of X. 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. Similarly, if every preimage of a base element of Y is open in X, then f is continuous.
• 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.

Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

See also

• Local base
• Subbase
• Gluing axiom