Productive Property

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 aleph_0 spaces. Separability is not productive, but is preserved by products of at most aleph_1 spaces.

Compactness is productive by the Tychonoff theorem.

See also

Hereditary Property, Divisible Property, Product Metric, Product Topology, Tychonoff Theorem

This entry contributed by Margherita Barile

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Joshi, K. D. "Productive Properties." §8.3 in Introduction to General Topology. New Delhi, India: Wiley, pp. 203-209, 1983.Kelley, J. L. General Topology. New York: Van Nostrand, p. 133, 1955.

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Productive Property

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Barile, Margherita. "Productive Property." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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