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In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. In Topology, the Cartesian product of Topological spaces can be given several different topologies However, the product topology is "correct" in that it makes the product space a pullback of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural". In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation

Contents

Definition

Let I be a (possibly infinite) index set and suppose Xi is a topological space for every i in I. In Mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index Set X = Π Xi, the Cartesian product of the sets Xi. For every i in I, we have a canonical projection pi : X → Xi. The product topology on X is defined to be the coarsest topology (i. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation e. the topology with the fewest open sets) for which all the projections pi are continuous. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function The product topology is sometimes called the Tychonoff topology.

Explicitly, the product topology on X can be described as the topology generated by sets of the form pi−1(U), where i in I and U is an open subset of Xi. In other words, the sets {pi−1(U)} form a subbase for the topology on X. In Highway engineering, subbase is a layer between Subgrade and the Base course. A subset of X is open if and only if it is a union of (possibly infinitely many) intersections of finitely many sets of the form pi−1(U). In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets 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 The pi−1(U) are sometimes called open cylinders, and their intersections are cylinder sets. In Mathematics, a cylinder set is the natural Open set of a Product topology. In Mathematics, a cylinder set is the natural Open set of a Product topology.

We can describe a basis for the product topology using bases of the constituting spaces Xi. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets A basis consists of sets \prod U_i, where for cofinitely many (all but finitely many) i, Ui = Xi (it's the whole space), and otherwise it's a basic open set of Xi. In Mathematics, a cofinite Subset of a set X is a subset Y whose complement in X is a finite set

In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the Xi gives a basis for the product \prod X_i.

In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In Topology, the Cartesian product of Topological spaces can be given several different topologies In general, the box topology is finer than the product topology, but for finite products they coincide. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation

Examples

If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn. 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

The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

Several additional examples are given in the article on the initial topology. In General topology and related areas of Mathematics, the initial topology ( projective topology or weak topology) on a set X

Properties

The product space X, together with the canonical projections, can be characterized by the following universal property: If Y is a topological space, and for every i in I, fi : Y → Xi is a continuous map, then there exists precisely one continuous map f : Y → X such that for each i in I the following diagram commutes:

Characteristic property of product spaces

This shows that the product space is a product in the category of topological spaces. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose If follows from the above universal property that a map f : Y → X is continuous iff fi = pi o f is continuous for all i in I. ↔ In many cases it is often easier to check that the component functions fi are continuous. Checking whether a map g : X→ Z is continuous is usually more difficult; one tries to use the fact that the pi are continuous in some way.

In addition to being continuous, the canonical projections pi : X → Xi are open maps. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets This means that any open subset of the product space remains open when projected down to the Xi. The converse is not true: if W is a subspace of the product space whose projections down to all the Xi are open, then W need not be open in X. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is (Consider for instance W = R2 \ (0,1)2. ) The canonical projections are not generally closed maps (consider for example the closed set \{(x,y) \in \mathbb{R}^2 \mid xy = 1\}, whose projections onto both axes are R \ {0}). In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets

The product topology is also called the topology of pointwise convergence because of the following fact: a sequence (or net) in X converges if and only if all its projections to the spaces Xi converge. In Mathematics, a sequence is an ordered list of objects (or events This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In particular, if one considers the space X = RI of all real valued functions on I, convergence in the product topology is the same as pointwise convergence of functions. In Mathematics, the real numbers may be described informally in several different ways The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

Any product of closed subsets of Xi is a closed set in X.

An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact. In Mathematics, Tychonoff's theorem states that the product of any collection of compact Topological spaces is compact This is easy to show for finite products, while the general statement is equivalent to the axiom of choice. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

Relation to other topological notions

A map that "locally looks like" a canonical projection F × U → U is called a fiber bundle. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space.

Axiom of choice

The axiom of choice is equivalent to the statement that the product of a non-empty collection of non-empty sets is non-empty. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. The proof is easy enough: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.

The axiom of choice occurs more generally in product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice. In Mathematics, Tychonoff's theorem states that the product of any collection of compact Topological spaces is compact

See also

References

  1. ^ Product topology preserves the Hausdorff property on PlanetMath
PlanetMath is a free, collaborative online Mathematics Encyclopedia.
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