Direct Product - Topological Space Direct Product

Topological Space Direct Product

The direct product for a collection of topological spaces Xi for i in I, some index set, once again makes use of the cartesian product

Defining the topology is a little tricky. For finitely many factors, this is the obvious and natural thing to do: simply take as a basis of open sets to be the collection of all cartesian products of open subsets from each factor:

This topology is called the product topology. For example, directly defining the product topology on R2 by the open sets of R (disjoint unions of open intervals), the basis for this topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usual metric topology).

The product topology for infinite products has a twist, and this 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 (i.e. to satisfy the categorical definition of product: the morphisms here are continuous functions): we take as a basis of open sets 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:

The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, and this does yield 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 which makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is only guaranteed to be open 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 the separate entry product topology.

Read more about this topic:  Direct Product

Famous quotes containing the words space, direct and/or product:

    The merit of those who fill a space in the world’s history, who are borne forward, as it were, by the weight of thousands whom they lead, shed a perfume less sweet than do the sacrifices of private virtue.
    Ralph Waldo Emerson (1803–1882)

    However strongly they resist it, our kids have to learn that as adults we need the companionship and love of other adults. The more direct we are about our needs, the easier it may be for our children to accept those needs. Their jealousy may come from a fear that if we adults love each other we might not have any left for them. We have to let them know that it’s a different kind of love.
    —Ruth Davidson Bell. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 3 (1978)

    Out of the thousand writers huffing and puffing through movieland there are scarcely fifty men and women of wit or talent. The rest of the fraternity is deadwood. Yet, in a curious way, there is not much difference between the product of a good writer and a bad one. They both have to toe the same mark.
    Ben Hecht (1893–1964)