Editing General topology (section)

==Products of spaces==
{{Main|Product topology}}
Given ''X'' such that

:<math>X := \prod_{i \in I} X_i,</math>

is the Cartesian product of the topological spaces ''X<sub>i</sub>'', [[index set|indexed]] by <math>i \in I</math>, and the '''[[projection (set theory)|canonical projections]]''' ''p<sub>i</sub>'' : ''X'' &rarr; ''X<sub>i</sub>'', the '''product topology''' on ''X'' is defined as the [[coarsest topology]] (i.e. the topology with the fewest open sets) for which all the projections ''p<sub>i</sub>'' are [[continuous (topology)|continuous]].  The product topology is sometimes called the '''Tychonoff topology'''.

The open sets in the product topology are unions (finite or infinite) of sets of the form <math>\prod_{i\in I} U_i</math>, where each ''U<sub>i</sub>'' is open in ''X<sub>i</sub>'' and ''U''<sub>''i''</sub>&nbsp;≠&nbsp;''X''<sub>''i''</sub> only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''X<sub>i</sub>'' gives a basis for the product <math>\prod_{i\in I} X_i</math>.

The product topology on ''X'' is the topology generated by sets of the form ''p<sub>i</sub>''<sup>&minus;1</sup>(''U''), where  ''i'' is in ''I ''  and ''U'' is an open subset of ''X<sub>i</sub>''. In other words, the sets {''p<sub>i</sub>''<sup>&minus;1</sup>(''U'')} form a [[subbase]] for the topology on ''X''. A [[subset]] of ''X'' is open if and only if it is a (possibly infinite) [[union (set theory)|union]] of [[intersection (set theory)|intersections]] of finitely many sets of the form ''p<sub>i</sub>''<sup>&minus;1</sup>(''U''). The ''p<sub>i</sub>''<sup>&minus;1</sup>(''U'') are sometimes called [[open cylinder]]s, and their intersections are [[cylinder set]]s.

In general, the product of the topologies of each ''X<sub>i</sub>'' forms a basis for what is called the [[box topology]] on ''X''. In general, the box topology is [[finer topology|finer]] than the product topology, but for finite products they coincide.

Related to compactness is [[Tychonoff's theorem]]: the (arbitrary) [[product topology|product]] of compact spaces is compact.