Editing Box topology (section)

==Comparison with product topology==

The basis sets in the product topology have almost the same definition as the above, ''except'' with the qualification that ''all but finitely many'' ''U<sub>i</sub>'' are equal to the component space ''X<sub>i</sub>''.  The product topology satisfies a very desirable property for maps ''f<sub>i</sub>'' : ''Y'' → ''X<sub>i</sub>'' into the component spaces: the product map  ''f'': ''Y'' → ''X'' defined by the component functions ''f<sub>i</sub>'' is [[continuous function (topology)|continuous]] if and only if all the ''f<sub>i</sub>'' are continuous.  As shown above, this does not always hold in the box topology. This actually makes the box topology very useful for providing [[counterexample]]s&mdash;many qualities such as [[compact space|compactness]], [[connected space|connectedness]], metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology. <!-- (''More specific examples here would be useful...'') -->