Editing Product topology (section)

==Relation to other topological notions==

'''Separation'''
* Every product of [[T0 space|T<sub>0</sub> space]]s is T<sub>0</sub>.
* Every product of [[T1 space|T<sub>1</sub> space]]s is T<sub>1</sub>.
* Every product of [[Hausdorff space]]s is Hausdorff.
* Every product of [[regular space]]s is regular.
* Every product of [[Tychonoff space]]s is Tychonoff.
* A product of [[normal space]]s {{em|need not}} be normal.

'''Compactness'''
* Every product of compact spaces is compact ([[Tychonoff's theorem]]).
* A product of [[locally compact space]]s {{em|need not}} be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact {{em|is}} locally compact (This condition is sufficient and necessary).

'''Connectedness'''
* Every product of [[Connectedness|connected]] (resp. path-connected) spaces is connected (resp. path-connected).
* Every product of hereditarily disconnected spaces is hereditarily disconnected.

'''Metric spaces'''
* Countable products of [[metric space]]s are [[metrizable space]]s.