Product topology

In topology, the cartesian product of topological spaces is turned into a topological space in the following way. Let I be a (possibly infinite) index set and suppose X_i is a topological space for every i in I. Set X = Π X_i, the cartesian product of the sets X_i. For every i in I, we have a canonical projection p_i : X -> X_i. The product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) which turns all the maps p_i into continuous maps.

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 in a simple way using the 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 element in that space, and let B be the product of the Y_i. Then, as long as X_i = Y_i, that is, we choose the entire space, for all but finitely many i in I, B will be a basis element of the product space, and a complete basis is generated in this way. In particular, this means that a finite product of spaces X has a simple basis given by products of bases in the X_i.

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.

Properties

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 X_i converge. In particular, if one considers the space X = R^I of all real valued functions on I, convergence in the product topology is the same as pointwise convergence of functions.

An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces 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 choice 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.

Relation to other topological notions

• Separation
  • Every product of T₀ spacess is T₀
  • Every product of T₁ spacess is T₁
  • Every product of Hausdorff spaces is Hausdorff
  • Every product of Regular spaces is Regular
  • Every product of Tychonoff spaces is Tychonoff
  • A product of normal spaces need not be normal
• Compactness
  • Every product of compact spaces is compact (Tychonoff's theorem)
  • A product of locally compact spaces need not be locally compact
• Connectedness
  • Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)
  • Every product of hereditarily disconnected spaces is hereditarily disconnected.

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