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:
“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)
“You will find that reason, which always ought to direct mankind, seldom does; but that passions and weaknesses commonly usurp its seat, and rule in its stead.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Man’s main task in life is to give birth to himself, to become what he potentially is. The most important product of his effort is his own personality.”
—Erich Fromm (1900–1980)