Regular space

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A topological space in which for every point and every closed set not containing there are open disjoint sets and such that and . A completely-regular space and, in particular, a metric space are regular.

If all one-point subsets in a regular space are closed (and this is not always true!), then the space is called a -space. Not every regular space is completely regular: there is an infinite -space on which every continuous real-valued function is constant. Moreover, not every regular space is normal (cf. Normal space). However, if a space is regular and each of its open coverings contains a countable subcovering, then it is normal. A space with a countable base is metrizable if and only if it is a -space. Regularity is inherited by any subspace and is multiplicative.


[1] J.L. Kelley, "General topology" , Springer (1975)
[2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)


See also Separation axiom for the hierarchy of . A topological property is said to be multiplicative if the product space has it if both and have the property. This is not to be confused with a "multiplicative system of subsetsmultiplicative system of subsets" , a phrase that is sometimes used to denote a collection of subsets that is closed under finite intersections.


[a1] E. Čech, "Topological spaces" , Wiley (1966) pp. 492ff
How to Cite This Entry:
Regular space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article