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Regular space

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A topological space in which for every point $x$ and every closed set $A$ not containing $x$ there are open disjoint sets $U$ and $V$ such that $x\in U$ and $A\subseteq V$. 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 $T_3$-space. Not every regular space is completely regular: there is an infinite $T_3$-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 $T_3$-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)


Comments

See also Separation axiom for the hierarchy of $T_0,T_1,\ldots$. A topological property is said to be multiplicative if the product space $X\times Y$ has it if both $X$ and $Y$ have the property. This is not to be confused with a "multiplicative system of subsets" , a phrase that is sometimes used to denote a collection of subsets that is closed under finite intersections.

References

[a1] E. Čech, "Topological spaces" , Wiley (1966) pp. 492ff
How to Cite This Entry:
Regular space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regular_space&oldid=31738
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