Product topology

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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. 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. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural".

Contents

Definition

Given X such that

or the (possibly infinite) Cartesian product of the topological spaces Xi, indexed by i \in I, and the canonical projections pi : XXi, the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.

The open sets in the product topology are unions (finite or infinite) of sets of the form \prod U_i, where each Ui is open in Xi and Ui≠Xi only finitely many times.

The product topology on X is 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. A subset of X is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form pi−1(U). The pi−1(U) are sometimes called open cylinders, and their intersections are cylinder sets.

We can describe a basis for the product topology using bases of the constituting spaces Xi. 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.

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