Editing Direct product (section)

== Topological space direct product ==
The direct product for a collection of [[topological spaces]] <math>X_i</math> for <math>i</math> in <math>I,</math> some index set, once again makes use of the Cartesian product
<math display=block>\prod_{i \in I} X_i.</math>

Defining the [[topology]] is a little tricky. For finitely many factors, it is the obvious and natural thing to do: simply take as a [[basis (topology)|basis]] of open sets to be the collection of all Cartesian products of open subsets from each factor:
<math display=block>\mathcal B = \left\{U_1 \times \cdots \times U_n\ : \ U_i\ \mathrm{open\ in}\ X_i\right\}.</math>

That topology is called the [[product topology]]. For example, by directly defining the product topology on <math>\R^2</math> by the open sets of <math>\R</math> (disjoint unions of open intervals), the basis for that topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usual [[metric space|metric]] topology).

The product topology for infinite products has a twist, which has to do with being able to make all the projection maps continuous and to make all functions into the product  continuous if and only if all its component functions are continuous (that is, to satisfy the categorical definition of product: the morphisms here are continuous functions). The basis of open sets is taken to be the collection of all Cartesian products of open subsets from each factor, as before, with the proviso that all but finitely many of the open subsets are the entire factor:
<math display=block>\mathcal B = \left\{ \prod_{i \in I} U_i\ : \ (\exists j_1,\ldots,j_n)(U_{j_i}\ \mathrm{open\ in}\ X_{j_i})\ \mathrm{and}\ (\forall i \neq j_1,\ldots,j_n)(U_i = X_i) \right\}.</math>

The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, which yields a somewhat interesting topology, the [[box topology]]. However, it is not too difficult to find an example of bunch of continuous component functions whose product function is not continuous (see the separate entry box topology for an example and more). The problem that makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is guaranteed to be open only for finitely many sets in the definition of topology.

Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff, the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called [[Tychonoff's theorem]], is yet another equivalence to the [[axiom of choice]].

For more properties and equivalent formulations, see [[product topology]].