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Difference between revisions of "Regular space"

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A [[Topological space|topological space]] in which for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808801.png" /> and every closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808802.png" /> not containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808803.png" /> there are open disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808805.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808807.png" />. A [[Completely-regular space|completely-regular space]] and, in particular, a [[Metric space|metric space]] are regular.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r0808809.png" />-space. Not every regular space is completely regular: there is an infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r08088010.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r08088011.png" />-space. Regularity is inherited by any subspace and is multiplicative.
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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====
 
====References====
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====Comments====
See also [[Separation axiom|Separation axiom]] for the hierarchy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r08088012.png" />. A topological property is said to be multiplicative if the product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r08088013.png" /> has it if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r08088014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080880/r08088015.png" /> 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.
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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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,  "Topological spaces" , Wiley  (1966)  pp. 492ff</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,  "Topological spaces" , Wiley  (1966)  pp. 492ff</TD></TR></table>

Latest revision as of 15:35, 15 April 2014

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://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