products of compact pavings are compact
Then, the product paving is defined by
Let be compact paved spaces for . Then, is a compact paving on .
Note that this result is a version of Tychonoff’s theorem applying to paved spaces and, together with the fact that all compact pavings are closed subsets of a compact space, is easily seen to be equivalent to Tychonoff’s theorem.
for any . By the finite intersection property, this is nonempty whenever is finite, so . Consequently, satisfies the finite intersection property and, by compactness of , the intersection is nonempty. So equation (1) shows that is nonempty.
|Title||products of compact pavings are compact|
|Date of creation||2013-03-22 18:45:09|
|Last modified on||2013-03-22 18:45:09|
|Last modified by||gel (22282)|