Home > Product topology

Explicitly, the topology on X can be described as follows. 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 p_i^-1(O), where i in I and O is an open subset of X_i. This implies that, in general, not all products of open sets need to be open in X.

We can describe a basis for the product topology using bases of the constituting spaces X_i. Suppose that for each i in I we choose a set Y_i which is either the whole space X_i or a basis set of that space, in such a way that X_i = Y_i for all but finitely many i in I. Let B be the cartesian product of the sets Y_i. The collection of all sets B that can be constructed in this fashion is a basis of the product space. In particular, this means that a product of finitely many spaces has a basis given by the products of base elements of the X_i.

1 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.

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.

2 Properties

The product topology is also called the topology of pointwise convergence because of the following fact: a sequenceThis is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or (or netTopology In mathematics the term net has at least two meanings. See the glossary of Riemannian and metric geometry for its meaning for metric spaces. This article is about its meaning in topology, where the concept of a net is a generalization of that of) in X converges if and only if all its projections to the spaces X_i converge. In particular, if one considers the space X = R^I of all realIn 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. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may valued functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtus on I, convergence in the product topology is the same as pointwise convergence of functions.

In addition to being continuous, the canonical projections p_i : X -> X_i are open maps. This means that any open subset of the product space remains open when projected down to the X_i. The converse is not true: if W is a subset of the product space whose projections down to all the X_i are open, then W need not be open in X. (Consider for instance W = R² \ (0,1)².)

An important theorem about the product topology is Tychonoff's theoremIn mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. For finite collections of compact spaces, this is not very surprising. The statement is in fact true for infinite collections of arbitr: any product of compact spacesIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e is compact. This is easy for finite products, but the statement is (surprisingly) also true for infinite products, when the proof requires the axiom of choiceSet theory In mathematics, the axiom of choice is an axiom in set theory. It was formulated about a century ago by Ernst Zermelo, and was quite controversial at the time. It states the following: Stated more formally: Another formulation of the axiom of c in some form.

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, f_i : Y -> X_i is a continuous map, then there exists precisely one continuous map f : Y -> X such that p_i o f = f_i for all i in I. This shows that the product space is a product in the sense of category theory.

To check whether a given map f : Y -> X is continuous, one can use the following handy criterion: f is continuous if and only if p_i o f is continuous for all i in I. In other words, if we write f as a tuple of its components, f=(f_i)_{i in I}, then f is continuous if and only if each of the f_i is. Checking whether a map g : X -> Z is continuous is usually more difficult; one tries to use the fact that the p_i are continuous in some way.