A property that is always fulfilled by the product of topological spaces, if it is fulfilled by each single factor. Examples of productive properties are connectedness, and path-connectedness, axioms T_0, T_1, T_2 and T_3, regularity and complete regularity, the property of being a Tychonoff space, but not axiom T_4 and normality, which does not even pass, in general, from a space X to X×X. Metrizability is not productive, but is preserved by products of at most ℵ_0 spaces. Separability is not productive, but is preserved by products of at most ℵ_1 spaces. Compactness is productive by the Tychonoff theorem.

divisible property | hereditary property | product metric | product topology | Tychonoff theorem