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.

References

 [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)