Editing Product topology (section)

==Axiom of choice==

One of many ways to express the [[axiom of choice]] is to say that it is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.<ref>{{citation|first=William J.|last=Pervin|title=Foundations of General Topology|year=1964|publisher=Academic Press|page=33}}</ref> The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.

The axiom of choice occurs again in the study of (topological) product spaces; for example, [[Tychonoff's theorem]] on compact sets is a more complex and subtle example of a statement that requires the axiom of choice and is equivalent to it in its most general formulation,<ref>{{citation|first1=John G.|last1=Hocking|first2=Gail S.|last2=Young|title=Topology|year=1988|orig-year=1961|publisher=Dover|isbn=978-0-486-65676-2|page=[https://archive.org/details/topology00hock_0/page/28 28]|url=https://archive.org/details/topology00hock_0/page/28}}</ref> and shows why the product topology may be considered the more useful topology to put on a Cartesian product.