Tychonoff product (named for Andrey Tikhonov) is simply the ? product in the category Top of topological spaces and continuous maps.
It is the usual notion of ‘product’ of topological spaces, but should be distinguished from the
box product, which is sometimes useful. (They are the same for ? finite products.) Definitions
be a ( X i ) i family of topological spaces. Consider the cartesian product of the ∏ i ∣ X i ∣ underlying sets of these spaces. This set comes with projection maps ? . π i : ∏ j ∣ X j ∣ → X i
Tychonoff topology or product topology on is the ∏ i ∣ X i ∣ initial topology generated by the . The π i Tychonoff product or topological product or simply product of the spaces is the set X i equipped with the Tychonoff product topology. ∏ i ∣ X i ∣ Properties
Of course, the maps
are π i continuous maps, so we have a cone in Top.
This is in fact a
product cone in . Top
Created on September 9, 2012 23:33:05