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A [[Regular space|regular space]] in which two disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766101.png" />-sets have disjoint neighbourhoods. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766102.png" />-space in which any two disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766103.png" />-sets have disjoint neighbourhoods is a quasi-normal space. Only for the quasi-normal spaces does the [[Stone–Čech compactification|Stone–Čech compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766104.png" /> coincide with the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766105.png" />. The following theorem provides a large supply of non-normal quasi-normal spaces: The product of any number of separated metric spaces is quasi-normal.
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A [[Regular space|regular space]] in which two disjoint $  \pi $-
 +
sets have disjoint neighbourhoods. Every $  T _  \lambda  $-
 +
space in which any two disjoint $  \pi $-
 +
sets have disjoint neighbourhoods is a quasi-normal space. Only for the quasi-normal spaces does the [[Stone–Čech compactification|Stone–Čech compactification]] $  \beta X $
 +
coincide with the space $  \omega _  \kappa  X $.  
 +
The following theorem provides a large supply of non-normal quasi-normal spaces: The product of any number of separated metric spaces is quasi-normal.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Zaitsev,  "Projection spectra"  ''Trans. Moscow Math. Soc.'' , '''27'''  (1972)  pp. 135–199  ''Trudy Moskov. Mat. Obshch.'' , '''27'''  (1972)  pp. 129–193</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.V. Shchepin,  "On the bicompact Ponomarev–Zaicev extension and the so-called spectral parasite"  ''Math. USSR Sb.'' , '''17'''  (1972)  pp. 317–326  ''Mat. Sb.'' , '''88''' :  2  (1972)  pp. 316–325</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Zaitsev,  "Projection spectra"  ''Trans. Moscow Math. Soc.'' , '''27'''  (1972)  pp. 135–199  ''Trudy Moskov. Mat. Obshch.'' , '''27'''  (1972)  pp. 129–193</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.V. Shchepin,  "On the bicompact Ponomarev–Zaicev extension and the so-called spectral parasite"  ''Math. USSR Sb.'' , '''17'''  (1972)  pp. 317–326  ''Mat. Sb.'' , '''88''' :  2  (1972)  pp. 316–325</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Quasi-normal spaces arose in the study of the spectrum of a topological space (cf. also [[Spectrum of spaces|Spectrum of spaces]]). This spectrum is obtained as follows. A partition of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766106.png" /> is a finite collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766107.png" /> of canonical closed sets (cf. [[Canonical set|Canonical set]]) that covers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766108.png" /> and the elements of which have disjoint interiors. The set of all these partitions is partially ordered by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q0766109.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661010.png" /> refines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661011.png" />. The nerve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661013.png" /> (cf. [[Nerve of a family of sets|Nerve of a family of sets]]) is the complex of subfamilies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661014.png" /> that have a non-empty intersection. There is an obvious simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661015.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661016.png" />. If the set of partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661017.png" /> is (upward) directed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661018.png" />, then the inverse spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661019.png" /> is the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661020.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661021.png" />. To obtain a suitable limit of this spectrum one takes the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661022.png" /> of maximal threads of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661023.png" />. A thread is a choice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661024.png" /> of simplexes with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661026.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661027.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661028.png" />. A thread <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661029.png" /> is maximal if whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661030.png" /> is another thread such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661031.png" /> is a face of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661032.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661033.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661034.png" />. The basic open sets are the sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661035.png" />.
+
Quasi-normal spaces arose in the study of the spectrum of a topological space (cf. also [[Spectrum of spaces|Spectrum of spaces]]). This spectrum is obtained as follows. A partition of a space $  X $
 +
is a finite collection $  {\mathcal A} $
 +
of canonical closed sets (cf. [[Canonical set|Canonical set]]) that covers $  X $
 +
and the elements of which have disjoint interiors. The set of all these partitions is partially ordered by: $  {\mathcal A} \succ {\mathcal A}  ^  \prime  $
 +
if and only if $  {\mathcal A} $
 +
refines $  {\mathcal A}  ^  \prime  $.  
 +
The nerve $  N _  {\mathcal A}  $
 +
of $  {\mathcal A} $(
 +
cf. [[Nerve of a family of sets|Nerve of a family of sets]]) is the complex of subfamilies of $  {\mathcal A} $
 +
that have a non-empty intersection. There is an obvious simplicial mapping $  \omega _ { {\mathcal A}  ^  \prime  } ^  {\mathcal A}  : N _  {\mathcal A}  \rightarrow N _ { {\mathcal A}  ^  \prime  } $
 +
if $  {\mathcal A} \succ {\mathcal A}  ^  \prime  $.  
 +
If the set of partitions of $  X $
 +
is (upward) directed by $  \succ $,  
 +
then the inverse spectrum $  \{ {N _  {\mathcal A}  , \omega _ { {\mathcal A}  ^  \prime  } ^  {\mathcal A}  } : { {\mathcal A}  \textrm{ a  partition  of  }  X } \} $
 +
is the spectrum of $  X $
 +
and is denoted by $  S _ {X} $.  
 +
To obtain a suitable limit of this spectrum one takes the set $  \widetilde{S}  {} _ {X} $
 +
of maximal threads of $  S _ {X} $.  
 +
A thread is a choice $  \{ t _  {\mathcal A}  \} _  {\mathcal A}  $
 +
of simplexes with $  t _  {\mathcal A}  \in {\mathcal A} $
 +
for all $  {\mathcal A} $
 +
and such that $  \omega _ { {\mathcal A}  ^  \prime  } ^  {\mathcal A}  ( t _  {\mathcal A}  ) = {\mathcal A}  ^  \prime  $
 +
whenever $  {\mathcal A} \succ {\mathcal A}  ^  \prime  $.  
 +
A thread $  \mathbf t = \{ t _  {\mathcal A}  \} _  {\mathcal A}  $
 +
is maximal if whenever $  \mathbf t  ^  \prime  = \{ t _  {\mathcal A}    ^  \prime  \} _  {\mathcal A}  $
 +
is another thread such that $  t _  {\mathcal A}  $
 +
is a face of $  t _  {\mathcal A}    ^  \prime  $
 +
for every $  {\mathcal A} $,  
 +
one has $  \mathbf t = \mathbf t  ^  \prime  $.  
 +
The basic open sets are the sets of the form $  O( t _  {\mathcal A}  ) = \{ {\mathbf t  ^  \prime  \in \widetilde{S}  {} _ {X} } : {t _  {\mathcal A}    ^  \prime    \textrm{ is  a  face  of  } t _  {\mathcal A}  } \} $.
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661036.png" />, first introduced in [[#References|[a3]]], is the space of all maximal centred systems of canonical closed sets topologized in the usual way, i.e. by taking the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661037.png" /> as a base for the closed sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661039.png" /> is the set of maximal centred systems to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661040.png" /> belongs.
+
The space $  \omega _  \kappa  X $,  
 +
first introduced in [[#References|[a3]]], is the space of all maximal centred systems of canonical closed sets topologized in the usual way, i.e. by taking the collection $  \{ {A  ^ {+} } : {A \textrm{ a  canonical  closed  set  } } \} $
 +
as a base for the closed sets of $  \omega _  \kappa  X $,  
 +
where $  A  ^ {+} $
 +
is the set of maximal centred systems to which $  A $
 +
belongs.
  
It turns out that there is a natural homeomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661041.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661042.png" />. Thus, for quasi-normal spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661043.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661044.png" />.
+
It turns out that there is a natural homeomorphism from $  \omega _  \kappa  X $
 +
onto $  \widetilde{S}  {} _ {X} $.  
 +
Thus, for quasi-normal spaces $  X $
 +
one has $  \beta X = \widetilde{S}  {} _ {X} = \omega _  \kappa  X $.
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661046.png" />-set is a finite intersection of closures of open sets. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661048.png" />-space, first introduced in [[#References|[1]]], is a semi-regular (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661049.png" />-) space all open sets of which are unions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661050.png" />-sets. I.e., a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661051.png" />-space is a semi-regular (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661052.png" />-) space (the canonical open sets form a base for the topology) in which the canonical closed sets form a [[Net (of sets in a topological space)|net (of sets in a topological space)]], i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661053.png" /> is open and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661054.png" />, then there is a canonical closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076610/q07661056.png" />.
+
A $  \pi $-
 +
set is a finite intersection of closures of open sets. A $  T _  \lambda  $-
 +
space, first introduced in [[#References|[1]]], is a semi-regular ( $  T _ {1} $-)  
 +
space all open sets of which are unions of $  \pi $-
 +
sets. I.e., a $  T _  \lambda  $-
 +
space is a semi-regular ( $  T _ {1} $-)  
 +
space (the canonical open sets form a base for the topology) in which the canonical closed sets form a [[Net (of sets in a topological space)|net (of sets in a topological space)]], i.e. if $  O $
 +
is open and $  x \in O $,  
 +
then there is a canonical closed set $  A $
 +
such that $  x \in A \subseteq O $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Kurosh,  "Kombinatorischer Aufbau der bikompakten topologischen Räume"  ''Compositio Math.'' , '''2'''  (1935)  pp. 471–476</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Zaitsev,  "Finite spectra of topological spaces and their limit spaces"  ''Math. Ann.'' , '''179'''  (1968–1969)  pp. 153–174</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Ponomarev,  "Paracompacta: their projection spectra and continuous mappings"  ''Mat. Sb.'' , '''60 (102)'''  (1963)  pp. 89–119  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Kurosh,  "Kombinatorischer Aufbau der bikompakten topologischen Räume"  ''Compositio Math.'' , '''2'''  (1935)  pp. 471–476</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Zaitsev,  "Finite spectra of topological spaces and their limit spaces"  ''Math. Ann.'' , '''179'''  (1968–1969)  pp. 153–174</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Ponomarev,  "Paracompacta: their projection spectra and continuous mappings"  ''Mat. Sb.'' , '''60 (102)'''  (1963)  pp. 89–119  (In Russian)</TD></TR></table>

Latest revision as of 08:09, 6 June 2020


A regular space in which two disjoint $ \pi $- sets have disjoint neighbourhoods. Every $ T _ \lambda $- space in which any two disjoint $ \pi $- sets have disjoint neighbourhoods is a quasi-normal space. Only for the quasi-normal spaces does the Stone–Čech compactification $ \beta X $ coincide with the space $ \omega _ \kappa X $. The following theorem provides a large supply of non-normal quasi-normal spaces: The product of any number of separated metric spaces is quasi-normal.

References

[1] V.I. Zaitsev, "Projection spectra" Trans. Moscow Math. Soc. , 27 (1972) pp. 135–199 Trudy Moskov. Mat. Obshch. , 27 (1972) pp. 129–193
[2] E.V. Shchepin, "On the bicompact Ponomarev–Zaicev extension and the so-called spectral parasite" Math. USSR Sb. , 17 (1972) pp. 317–326 Mat. Sb. , 88 : 2 (1972) pp. 316–325

Comments

Quasi-normal spaces arose in the study of the spectrum of a topological space (cf. also Spectrum of spaces). This spectrum is obtained as follows. A partition of a space $ X $ is a finite collection $ {\mathcal A} $ of canonical closed sets (cf. Canonical set) that covers $ X $ and the elements of which have disjoint interiors. The set of all these partitions is partially ordered by: $ {\mathcal A} \succ {\mathcal A} ^ \prime $ if and only if $ {\mathcal A} $ refines $ {\mathcal A} ^ \prime $. The nerve $ N _ {\mathcal A} $ of $ {\mathcal A} $( cf. Nerve of a family of sets) is the complex of subfamilies of $ {\mathcal A} $ that have a non-empty intersection. There is an obvious simplicial mapping $ \omega _ { {\mathcal A} ^ \prime } ^ {\mathcal A} : N _ {\mathcal A} \rightarrow N _ { {\mathcal A} ^ \prime } $ if $ {\mathcal A} \succ {\mathcal A} ^ \prime $. If the set of partitions of $ X $ is (upward) directed by $ \succ $, then the inverse spectrum $ \{ {N _ {\mathcal A} , \omega _ { {\mathcal A} ^ \prime } ^ {\mathcal A} } : { {\mathcal A} \textrm{ a partition of } X } \} $ is the spectrum of $ X $ and is denoted by $ S _ {X} $. To obtain a suitable limit of this spectrum one takes the set $ \widetilde{S} {} _ {X} $ of maximal threads of $ S _ {X} $. A thread is a choice $ \{ t _ {\mathcal A} \} _ {\mathcal A} $ of simplexes with $ t _ {\mathcal A} \in {\mathcal A} $ for all $ {\mathcal A} $ and such that $ \omega _ { {\mathcal A} ^ \prime } ^ {\mathcal A} ( t _ {\mathcal A} ) = {\mathcal A} ^ \prime $ whenever $ {\mathcal A} \succ {\mathcal A} ^ \prime $. A thread $ \mathbf t = \{ t _ {\mathcal A} \} _ {\mathcal A} $ is maximal if whenever $ \mathbf t ^ \prime = \{ t _ {\mathcal A} ^ \prime \} _ {\mathcal A} $ is another thread such that $ t _ {\mathcal A} $ is a face of $ t _ {\mathcal A} ^ \prime $ for every $ {\mathcal A} $, one has $ \mathbf t = \mathbf t ^ \prime $. The basic open sets are the sets of the form $ O( t _ {\mathcal A} ) = \{ {\mathbf t ^ \prime \in \widetilde{S} {} _ {X} } : {t _ {\mathcal A} ^ \prime \textrm{ is a face of } t _ {\mathcal A} } \} $.

The space $ \omega _ \kappa X $, first introduced in [a3], is the space of all maximal centred systems of canonical closed sets topologized in the usual way, i.e. by taking the collection $ \{ {A ^ {+} } : {A \textrm{ a canonical closed set } } \} $ as a base for the closed sets of $ \omega _ \kappa X $, where $ A ^ {+} $ is the set of maximal centred systems to which $ A $ belongs.

It turns out that there is a natural homeomorphism from $ \omega _ \kappa X $ onto $ \widetilde{S} {} _ {X} $. Thus, for quasi-normal spaces $ X $ one has $ \beta X = \widetilde{S} {} _ {X} = \omega _ \kappa X $.

A $ \pi $- set is a finite intersection of closures of open sets. A $ T _ \lambda $- space, first introduced in [1], is a semi-regular ( $ T _ {1} $-) space all open sets of which are unions of $ \pi $- sets. I.e., a $ T _ \lambda $- space is a semi-regular ( $ T _ {1} $-) space (the canonical open sets form a base for the topology) in which the canonical closed sets form a net (of sets in a topological space), i.e. if $ O $ is open and $ x \in O $, then there is a canonical closed set $ A $ such that $ x \in A \subseteq O $.

References

[a1] A. Kurosh, "Kombinatorischer Aufbau der bikompakten topologischen Räume" Compositio Math. , 2 (1935) pp. 471–476
[a2] V.I. Zaitsev, "Finite spectra of topological spaces and their limit spaces" Math. Ann. , 179 (1968–1969) pp. 153–174
[a3] V.I. Ponomarev, "Paracompacta: their projection spectra and continuous mappings" Mat. Sb. , 60 (102) (1963) pp. 89–119 (In Russian)
How to Cite This Entry:
Quasi-normal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-normal_space&oldid=18850
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article