View source for Regular space

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A [[Topological space|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|completely-regular space]] and, in particular, a [[Metric space|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>



====Comments====
See also [[Separation axiom|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,   "Topological spaces" , Wiley  (1966)  pp. 492ff</TD></TR></table>