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The absolute of a regular topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102901.png" /> is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102902.png" /> which is mapped perfectly and irreducibly onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102903.png" />, and is such that any perfect irreducible inverse image of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102904.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102905.png" />. Each regular space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102906.png" /> has a unique absolute. The absolute of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102907.png" /> is always extremally disconnected and completely regular, and is perfectly and irreducibly mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102908.png" /> by means of a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a0102909.png" />. If two spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029011.png" /> are connected by a single-valued or multi-valued [[Perfect irreducible mapping|perfect irreducible mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029012.png" />, then their absolutes are homeomorphic, and there exists a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029014.png" />.
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If a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029015.png" /> is given, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029016.png" /> will be, in the general case, multi-valued, irreducible and perfect. Thus, absolutes and their homeomorphisms  "control"  the entire class of perfect irreducible mappings of regular spaces. The meaning of this fundamental property is that absolutes of regular topological spaces are projective objects in the category of regular spaces and perfect irreducible mappings. If a regular space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029017.png" /> is compact, respectively, finally compact or complete in the sense of Čech, the respective property is also displayed by the absolute of this space. The absolute of a paracompact space is even strongly paracompact, and is, moreover, perfectly zero-dimensional. However, the absolute of a normal space need not itself be normal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029018.png" /> is a completely regular space, then the [[Stone–Čech compactification|Stone–Čech compactification]] of its absolute is the absolute of any compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029019.png" />. Two spaces are called co-absolute if their absolutes are homeomorphic.
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{{TEX|done}}
  
Thus, the class of regular spaces is subdivided into disjoint (pairwise not intersecting) classes of co-absolute spaces. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029020.png" /> is co-absolute with some metric space if and only if it is a paracompact feathered space containing a dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029021.png" />-discrete system of open sets. A compact space is co-absolute with some metrizable compactum if and only if it has a countable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029022.png" />-weight. If a compact space has a countable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029023.png" />-weight and has no isolated points, then and only then it will be co-absolute with the perfect Cantor set. Consequently, all metrizable compacta without isolated points are co-absolute with the perfect Cantor set. The absolute of a countable metrizable compactum is an extension of the Stone–Čech compactification of the natural numbers. The absolute of an extremally disconnected space is homeomorphic to it. Thus, the class of absolutes (whatever this may be) of regular spaces coincides with the class of extremally disconnected spaces. Since a non-discrete extremally disconnected space does not contain any convergent sequence of pairwise distinct points, the absolute of any non-discrete space is non-metrizable (and does not even satisfy the first axiom of countability).
+
The absolute of a regular topological space  $  X $
 +
is the space $  aX $
 +
which is mapped perfectly and irreducibly onto  $  X $,
 +
and is such that any perfect irreducible inverse image of the space $  X $
 +
is homeomorphic to  $  aX $.  
 +
Each regular space $  X $
 +
has a unique absolute. The absolute of a space  $  X $
 +
is always extremally disconnected and completely regular, and is perfectly and irreducibly mapped onto  $  X $
 +
by means of a transformation  $  \pi _ {X} :  aX \rightarrow X $.  
 +
If two spaces $  X $
 +
and  $  Y $
 +
are connected by a single-valued or multi-valued [[Perfect irreducible mapping|perfect irreducible mapping]]  $  f:  X \rightarrow Y $,
 +
then their absolutes are homeomorphic, and there exists a homeomorphism  $  f _ {a} :  aX \rightarrow aY $
 +
such that  $  f = \pi _ {Y} f _ {a} \pi _ {X}  ^ {-1} $.
  
Of the numerous ways in which the absolute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029024.png" /> of a given (regular) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029025.png" /> can be constructed, the following method is one of the simplest.
+
If a homeomorphism  $  f _ {a} :  aX \rightarrow aY $
 +
is given, the mapping  $  f = \pi _ {Y} f _ {a} \pi _ {X}  ^ {-1} $
 +
will be, in the general case, multi-valued, irreducible and perfect. Thus, absolutes and their homeomorphisms  "control" the entire class of perfect irreducible mappings of regular spaces. The meaning of this fundamental property is that absolutes of regular topological spaces are projective objects in the category of regular spaces and perfect irreducible mappings. If a regular space  $  X $
 +
is compact, respectively, finally compact or complete in the sense of Čech, the respective property is also displayed by the absolute of this space. The absolute of a paracompact space is even strongly paracompact, and is, moreover, perfectly zero-dimensional. However, the absolute of a normal space need not itself be normal. If  $  X $
 +
is a completely regular space, then the [[Stone–Čech compactification|Stone–Čech compactification]] of its absolute is the absolute of any compactification of  $  X $.
 +
Two spaces are called co-absolute if their absolutes are homeomorphic.
  
A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029026.png" /> of non-empty canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029027.png" />-sets, i.e. of canonical closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029028.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029029.png" />, is called a thread if it is inclusion-directed, i.e. if for each two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029030.png" /> of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029031.png" /> there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029032.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029033.png" />. A thread <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029034.png" /> is called a maximal or an end thread if it is not a subfamily of any thread different from it. It can be shown that threads exist; moreover, it can be shown that for each non-empty <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029035.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029036.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029037.png" /> of all threads containing the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029038.png" /> as an element is non-empty. Each thread is contained in some maximal thread. The intersection of all sets which are elements of a maximal thread <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029039.png" /> is either empty or contains a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029040.png" />; in the latter case the thread <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029041.png" /> is said to be convergent (to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029042.png" />). A topology is introduced in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029043.png" /> of all ends (i.e. maximal threads), by taking the collection of all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029044.png" /> as a basis for the closed sets. The resulting topology is Hausdorff and compact. The convergent ends in the compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029045.png" /> form an everywhere-dense subspace. The subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029046.png" /> consisting of the convergent ends is at the same time the absolute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029048.png" />; it turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029049.png" /> is identical with the Stone–Čech compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029052.png" /> is not only regular, but completely regular, the formula of commutativity of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029054.png" /> is valid:
+
Thus, the class of regular spaces is subdivided into disjoint (pairwise not intersecting) classes of co-absolute spaces. A space  $  X $
 +
is co-absolute with some metric space if and only if it is a paracompact feathered space containing a dense  $  \sigma $-
 +
discrete system of open sets. A compact space is co-absolute with some metrizable compactum if and only if it has a countable  $  \pi $-
 +
weight. If a compact space has a countable  $  \pi $-
 +
weight and has no isolated points, then and only then it will be co-absolute with the perfect Cantor set. Consequently, all metrizable compacta without isolated points are co-absolute with the perfect Cantor set. The absolute of a countable metrizable compactum is an extension of the Stone–Čech compactification of the natural numbers. The absolute of an extremally disconnected space is homeomorphic to it. Thus, the class of absolutes (whatever this may be) of regular spaces coincides with the class of extremally disconnected spaces. Since a non-discrete extremally disconnected space does not contain any convergent sequence of pairwise distinct points, the absolute of any non-discrete space is non-metrizable (and does not even satisfy the first axiom of countability).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029055.png" /></td> </tr></table>
+
Of the numerous ways in which the absolute  $  aX $
 +
of a given (regular) space  $  X $
 +
can be constructed, the following method is one of the simplest.
  
 +
A family  $  \xi = \{ A \} $
 +
of non-empty canonical  $  \kappa a $-
 +
sets, i.e. of canonical closed sets  $  A $
 +
of the space  $  X $,
 +
is called a thread if it is inclusion-directed, i.e. if for each two elements  $  A, A  ^  \prime  $
 +
of the family  $  \xi $
 +
there exists an element  $  A ^ {\prime\prime } $
 +
contained in  $  A \cap A  ^  \prime  $.
 +
A thread  $  \xi $
 +
is called a maximal or an end thread if it is not a subfamily of any thread different from it. It can be shown that threads exist; moreover, it can be shown that for each non-empty  $  \kappa a $-
 +
set  $  A $
 +
the set  $  D _ {A} $
 +
of all threads containing the set  $  A $
 +
as an element is non-empty. Each thread is contained in some maximal thread. The intersection of all sets which are elements of a maximal thread  $  \xi $
 +
is either empty or contains a single point  $  x ( \xi ) $;
 +
in the latter case the thread  $  \xi $
 +
is said to be convergent (to the point  $  x ( \xi ) $).
 +
A topology is introduced in the set  $  \overline{a}\; X $
 +
of all ends (i.e. maximal threads), by taking the collection of all sets  $  D _ {A} $
 +
as a basis for the closed sets. The resulting topology is Hausdorff and compact. The convergent ends in the compactum  $  \overline{a}\; X $
 +
form an everywhere-dense subspace. The subspace of the space  $  \overline{a}\; X $
 +
consisting of the convergent ends is at the same time the absolute  $  aX $
 +
of  $  X $;
 +
it turns out that  $  \overline{a}\; X $
 +
is identical with the Stone–Čech compactification  $  \beta a X $
 +
of  $  aX $.
 +
If  $  X $
 +
is not only regular, but completely regular, the formula of commutativity of the operators  $  a $
 +
and  $  \beta $
 +
is valid:
  
 +
$$
 +
\overline{a}\; X  =  \beta a X  =  a \beta X .
 +
$$
  
 
====Comments====
 
====Comments====
The definition of the absolute of a regular topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029056.png" /> given above is slightly imprecise. A better (more precise) definition is: The absolute of a regular topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029057.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029059.png" /> is a perfect irreducible mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029060.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029061.png" />, such that for every regular topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029062.png" /> and every perfect irreducible mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029063.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029064.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029065.png" /> there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029067.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029069.png" />.
+
The definition of the absolute of a regular topological space $  X $
 +
given above is slightly imprecise. A better (more precise) definition is: The absolute of a regular topological space $  X $
 +
is a pair $  ( a X , \pi _ {X} ) $,  
 +
where $  \pi _ {X} $
 +
is a perfect irreducible mapping of a X $
 +
onto $  X $,  
 +
such that for every regular topological space $  Y $
 +
and every perfect irreducible mapping $  f $
 +
of $  Y $
 +
onto $  X $
 +
there is a mapping $  g $
 +
of a X $
 +
onto $  Y $
 +
such that $  \pi _ {X} = f g $.
  
 
In Western literature finally compact spaces are also called Lindelöf spaces and canonical closed set are also called regular closed sets (or sometimes regularly closed sets). The maximal or end threads are commonly called regular closed ultrafilters.
 
In Western literature finally compact spaces are also called Lindelöf spaces and canonical closed set are also called regular closed sets (or sometimes regularly closed sets). The maximal or end threads are commonly called regular closed ultrafilters.
  
The simple construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029070.png" /> given above can be rephrased as follows: The family of regular closed sets forms, in a natural way, a complete [[Boolean algebra|Boolean algebra]]. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029071.png" /> is then simply the [[Stone space|Stone space]] of this Boolean algebra (the set of all ultrafilters (cf. [[Ultrafilter|Ultrafilter]]) on it, topologized using the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029072.png" /> as a [[Base|base]] for the closed sets).
+
The simple construction of $  \overline{a}\; X $
 +
given above can be rephrased as follows: The family of regular closed sets forms, in a natural way, a complete [[Boolean algebra|Boolean algebra]]. The space $  \overline{a}\; X $
 +
is then simply the [[Stone space|Stone space]] of this Boolean algebra (the set of all ultrafilters (cf. [[Ultrafilter|Ultrafilter]]) on it, topologized using the sets $  D _ {A} $
 +
as a [[Base|base]] for the closed sets).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</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>
 
<table><TR><TD valign="top">[a1]</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>
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029075.png" />-absolute of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029076.png" />-proximity space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029077.png" /> is the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029078.png" /> consisting of a [[Proximity space|proximity space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029079.png" /> and a projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029080.png" /> which is a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029081.png" />-mapping. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029082.png" />-mapping is a term denoting any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029083.png" />-perfect, irreducible, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029084.png" />-proximity-continuous mapping. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029085.png" />-proximity space has a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029086.png" />-absolute. Any regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029087.png" />-mapping on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029088.png" />-absolute is a proximity equivalence. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029089.png" />-absolute of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029090.png" /> is the maximal inverse image of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029091.png" /> under regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029092.png" />-mappings. For each regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029093.png" />-mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029094.png" /> there exists a proximity equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029095.png" /> such that the diagram
+
The $  \theta $-
 +
absolute of a $  \theta $-
 +
proximity space $  (X, \delta ) $
 +
is the pair $  (X _  \delta  , \pi _ {X} ) $
 +
consisting of a [[Proximity space|proximity space]] $  X _  \delta  $
 +
and a projection $  \pi _ {X} : X _  \delta  \rightarrow X $
 +
which is a regular $  \theta $-
 +
mapping. Here $  \theta $-
 +
mapping is a term denoting any $  \theta $-
 +
perfect, irreducible, $  \theta $-
 +
proximity-continuous mapping. Any $  \theta $-
 +
proximity space has a unique $  \theta $-
 +
absolute. Any regular $  \theta $-
 +
mapping on a $  \theta $-
 +
absolute is a proximity equivalence. The $  \theta $-
 +
absolute of a space $  ( X, \delta ) $
 +
is the maximal inverse image of the space $  (X, \delta ) $
 +
under regular $  \theta $-
 +
mappings. For each regular $  \theta $-
 +
mapping $  f: (X, \delta ) \rightarrow (Y, \delta  ^  \prime  ) $
 +
there exists a proximity equivalence $  F : X _  \delta  \rightarrow Y _ {\delta  ^  \prime  } $
 +
such that the diagram
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029096.png" /></td> </tr></table>
+
$$
 +
 
 +
\begin{array}{rcl}
 +
X _  \delta  & \rightarrow ^ { F }    &Y _ {\delta  ^  \prime  }  \\
 +
\pi _ {X} \downarrow  &{}  &\downarrow \pi _ {Y}  \\
 +
X  & \rightarrow _ { f }  & Y  \\
 +
\end{array}
 +
 
 +
$$
  
 
is commutative.
 
is commutative.
  
For maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029097.png" />-proximities on regular topological spaces the concept of a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029098.png" />-mapping is identical with that of a perfect irreducible mapping, while the concept of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a01029099.png" />-absolute is identical with that of the absolute of a regular topological space.
+
For maximal $  \theta $-
 +
proximities on regular topological spaces the concept of a regular $  \theta $-
 +
mapping is identical with that of a perfect irreducible mapping, while the concept of a $  \theta $-
 +
absolute is identical with that of the absolute of a regular topological space.
  
 
====References====
 
====References====
Line 36: Line 149:
 
''V.V. Fedorchuk''
 
''V.V. Fedorchuk''
  
The absolute in projective geometry is the curve (surface) of the second order constituting the set of infinitely-distant points in the [[Klein interpretation|Klein interpretation]] of a hyperbolic plane (space). The absolute can be used to introduce a metric on a projective plane (space) (cf. [[Projective determination of a metric|Projective determination of a metric]]). For instance, the projective measure of a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a010290100.png" /> is defined as a quantity which is proportional to the natural logarithm of the [[Double ratio|double ratio]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a010290101.png" /> of four points, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a010290102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a010290103.png" /> are the points of intersection of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010290/a010290104.png" /> with the absolute.
+
The absolute in projective geometry is the curve (surface) of the second order constituting the set of infinitely-distant points in the [[Klein interpretation|Klein interpretation]] of a hyperbolic plane (space). The absolute can be used to introduce a metric on a projective plane (space) (cf. [[Projective determination of a metric|Projective determination of a metric]]). For instance, the projective measure of a segment $  AB $
 +
is defined as a quantity which is proportional to the natural logarithm of the [[Double ratio|double ratio]] $  (ABCD) $
 +
of four points, where $  C $
 +
and $  D $
 +
are the points of intersection of the straight line $  AB $
 +
with the absolute.
  
 
''A.B. Ivanov''
 
''A.B. Ivanov''

Revision as of 16:08, 1 April 2020


The absolute of a regular topological space $ X $ is the space $ aX $ which is mapped perfectly and irreducibly onto $ X $, and is such that any perfect irreducible inverse image of the space $ X $ is homeomorphic to $ aX $. Each regular space $ X $ has a unique absolute. The absolute of a space $ X $ is always extremally disconnected and completely regular, and is perfectly and irreducibly mapped onto $ X $ by means of a transformation $ \pi _ {X} : aX \rightarrow X $. If two spaces $ X $ and $ Y $ are connected by a single-valued or multi-valued perfect irreducible mapping $ f: X \rightarrow Y $, then their absolutes are homeomorphic, and there exists a homeomorphism $ f _ {a} : aX \rightarrow aY $ such that $ f = \pi _ {Y} f _ {a} \pi _ {X} ^ {-1} $.

If a homeomorphism $ f _ {a} : aX \rightarrow aY $ is given, the mapping $ f = \pi _ {Y} f _ {a} \pi _ {X} ^ {-1} $ will be, in the general case, multi-valued, irreducible and perfect. Thus, absolutes and their homeomorphisms "control" the entire class of perfect irreducible mappings of regular spaces. The meaning of this fundamental property is that absolutes of regular topological spaces are projective objects in the category of regular spaces and perfect irreducible mappings. If a regular space $ X $ is compact, respectively, finally compact or complete in the sense of Čech, the respective property is also displayed by the absolute of this space. The absolute of a paracompact space is even strongly paracompact, and is, moreover, perfectly zero-dimensional. However, the absolute of a normal space need not itself be normal. If $ X $ is a completely regular space, then the Stone–Čech compactification of its absolute is the absolute of any compactification of $ X $. Two spaces are called co-absolute if their absolutes are homeomorphic.

Thus, the class of regular spaces is subdivided into disjoint (pairwise not intersecting) classes of co-absolute spaces. A space $ X $ is co-absolute with some metric space if and only if it is a paracompact feathered space containing a dense $ \sigma $- discrete system of open sets. A compact space is co-absolute with some metrizable compactum if and only if it has a countable $ \pi $- weight. If a compact space has a countable $ \pi $- weight and has no isolated points, then and only then it will be co-absolute with the perfect Cantor set. Consequently, all metrizable compacta without isolated points are co-absolute with the perfect Cantor set. The absolute of a countable metrizable compactum is an extension of the Stone–Čech compactification of the natural numbers. The absolute of an extremally disconnected space is homeomorphic to it. Thus, the class of absolutes (whatever this may be) of regular spaces coincides with the class of extremally disconnected spaces. Since a non-discrete extremally disconnected space does not contain any convergent sequence of pairwise distinct points, the absolute of any non-discrete space is non-metrizable (and does not even satisfy the first axiom of countability).

Of the numerous ways in which the absolute $ aX $ of a given (regular) space $ X $ can be constructed, the following method is one of the simplest.

A family $ \xi = \{ A \} $ of non-empty canonical $ \kappa a $- sets, i.e. of canonical closed sets $ A $ of the space $ X $, is called a thread if it is inclusion-directed, i.e. if for each two elements $ A, A ^ \prime $ of the family $ \xi $ there exists an element $ A ^ {\prime\prime } $ contained in $ A \cap A ^ \prime $. A thread $ \xi $ is called a maximal or an end thread if it is not a subfamily of any thread different from it. It can be shown that threads exist; moreover, it can be shown that for each non-empty $ \kappa a $- set $ A $ the set $ D _ {A} $ of all threads containing the set $ A $ as an element is non-empty. Each thread is contained in some maximal thread. The intersection of all sets which are elements of a maximal thread $ \xi $ is either empty or contains a single point $ x ( \xi ) $; in the latter case the thread $ \xi $ is said to be convergent (to the point $ x ( \xi ) $). A topology is introduced in the set $ \overline{a}\; X $ of all ends (i.e. maximal threads), by taking the collection of all sets $ D _ {A} $ as a basis for the closed sets. The resulting topology is Hausdorff and compact. The convergent ends in the compactum $ \overline{a}\; X $ form an everywhere-dense subspace. The subspace of the space $ \overline{a}\; X $ consisting of the convergent ends is at the same time the absolute $ aX $ of $ X $; it turns out that $ \overline{a}\; X $ is identical with the Stone–Čech compactification $ \beta a X $ of $ aX $. If $ X $ is not only regular, but completely regular, the formula of commutativity of the operators $ a $ and $ \beta $ is valid:

$$ \overline{a}\; X = \beta a X = a \beta X . $$

Comments

The definition of the absolute of a regular topological space $ X $ given above is slightly imprecise. A better (more precise) definition is: The absolute of a regular topological space $ X $ is a pair $ ( a X , \pi _ {X} ) $, where $ \pi _ {X} $ is a perfect irreducible mapping of $ a X $ onto $ X $, such that for every regular topological space $ Y $ and every perfect irreducible mapping $ f $ of $ Y $ onto $ X $ there is a mapping $ g $ of $ a X $ onto $ Y $ such that $ \pi _ {X} = f g $.

In Western literature finally compact spaces are also called Lindelöf spaces and canonical closed set are also called regular closed sets (or sometimes regularly closed sets). The maximal or end threads are commonly called regular closed ultrafilters.

The simple construction of $ \overline{a}\; X $ given above can be rephrased as follows: The family of regular closed sets forms, in a natural way, a complete Boolean algebra. The space $ \overline{a}\; X $ is then simply the Stone space of this Boolean algebra (the set of all ultrafilters (cf. Ultrafilter) on it, topologized using the sets $ D _ {A} $ as a base for the closed sets).

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

The $ \theta $- absolute of a $ \theta $- proximity space $ (X, \delta ) $ is the pair $ (X _ \delta , \pi _ {X} ) $ consisting of a proximity space $ X _ \delta $ and a projection $ \pi _ {X} : X _ \delta \rightarrow X $ which is a regular $ \theta $- mapping. Here $ \theta $- mapping is a term denoting any $ \theta $- perfect, irreducible, $ \theta $- proximity-continuous mapping. Any $ \theta $- proximity space has a unique $ \theta $- absolute. Any regular $ \theta $- mapping on a $ \theta $- absolute is a proximity equivalence. The $ \theta $- absolute of a space $ ( X, \delta ) $ is the maximal inverse image of the space $ (X, \delta ) $ under regular $ \theta $- mappings. For each regular $ \theta $- mapping $ f: (X, \delta ) \rightarrow (Y, \delta ^ \prime ) $ there exists a proximity equivalence $ F : X _ \delta \rightarrow Y _ {\delta ^ \prime } $ such that the diagram

$$ \begin{array}{rcl} X _ \delta & \rightarrow ^ { F } &Y _ {\delta ^ \prime } \\ \pi _ {X} \downarrow &{} &\downarrow \pi _ {Y} \\ X & \rightarrow _ { f } & Y \\ \end{array} $$

is commutative.

For maximal $ \theta $- proximities on regular topological spaces the concept of a regular $ \theta $- mapping is identical with that of a perfect irreducible mapping, while the concept of a $ \theta $- absolute is identical with that of the absolute of a regular topological space.

References

[1] V.I. Ponomarev, "On spaces co-abolute with metric space" Russian Math. Surveys , 21 : 4 (1966) pp. 87–114 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 101–132
[2] A.M. Gleason, "Projective topological spaces" Illinois J. Math. , 2 : 4A (1958) pp. 482–489
[3] V.I. Ponomarev, "Paracompacta, their projection spectra and continuous images" Mat. Sb. , 60 (102) : 1 (1963) pp. 89–119 (In Russian)
[4] V.V. Fedorchuk, "Perfect irreducible mappings and generalized proximities" Math. USSR-Sb. , 5 : 4 (1968) pp. 498–508 Mat. Sb. , 74 (118) : 4 (1968) pp. 513–536

V.V. Fedorchuk

The absolute in projective geometry is the curve (surface) of the second order constituting the set of infinitely-distant points in the Klein interpretation of a hyperbolic plane (space). The absolute can be used to introduce a metric on a projective plane (space) (cf. Projective determination of a metric). For instance, the projective measure of a segment $ AB $ is defined as a quantity which is proportional to the natural logarithm of the double ratio $ (ABCD) $ of four points, where $ C $ and $ D $ are the points of intersection of the straight line $ AB $ with the absolute.

A.B. Ivanov

How to Cite This Entry:
Absolute. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute&oldid=45002
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article