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The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d0315203.png" /> defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d0315204.png" />. The diagonal product of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d0315205.png" /> satisfies, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d0315206.png" />, the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d0315207.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d0315208.png" /> denotes the projection of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d0315209.png" /> on the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152010.png" />. The diagonal product of continuous mappings is continuous. A family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152011.png" /> of topological spaces is said to be partitioning if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152012.png" /> and neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152014.png" /> there exist an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152015.png" /> and an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152018.png" /> is a partitioning family of mappings and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152019.png" /> is the diagonal product of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152021.png" /> is an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152022.png" /> into the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152023.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152024.png" /> is a homeomorphism. The diagonal product of mappings was used by A.N. Tikhonov to imbed a completely-regular space of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152025.png" /> in the cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152026.png" />.
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'' $  f _  \alpha  :  X \rightarrow Y _  \alpha  $,
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$  \alpha \in {\mathcal A} $''
  
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The mapping  $  f:  X \rightarrow Y = \prod \{ {Y _  \alpha  } : {\alpha \in {\mathcal A} } \} $
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defined by the equation  $  f ( x) = \{ f _  \alpha  ( x) \} \in Y $.
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The diagonal product of mappings  $  f _  \alpha  $
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satisfies, for any  $  \alpha $,
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the relation  $  f _  \alpha  = \pi _  \alpha  f $,
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where  $  \pi _  \alpha  $
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denotes the projection of the product  $  Y $
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on the factor  $  Y _  \alpha  $.
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The diagonal product of continuous mappings is continuous. A family of mappings  $  f _  \alpha  :  X \rightarrow Y _  \alpha  $
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of topological spaces is said to be partitioning if for any point  $  x \in X $
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and neighbourhood  $  Ox $
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of  $  x $
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there exist an index  $  \alpha $
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and an open subset  $  U _  \alpha  \subset  Y _  \alpha  $
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such that  $  x \in f _  \alpha  ^ { - 1 } U _  \alpha  \subset  Ox $.
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If  $  \{ f _  \alpha  :  X \rightarrow Y _  \alpha  \} $
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is a partitioning family of mappings and if  $  f $
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is the diagonal product of the mappings  $  f _  \alpha  $,
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then  $  f $
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is an imbedding of  $  X $
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into the product  $  \prod Y _  \alpha  $,
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i.e.  $  f:  X \rightarrow fX $
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is a homeomorphism. The diagonal product of mappings was used by A.N. Tikhonov to imbed a completely-regular space of weight  $  \tau $
 +
in the cube  $  I ^ { \tau } $.
  
 
====Comments====
 
====Comments====
 
Instead of calling a family of mappings partitioning, one says that it separates points and closed sets.
 
Instead of calling a family of mappings partitioning, one says that it separates points and closed sets.
  
In an arbitrary category with products, cf. [[Direct product|Direct product]], the diagonal product of mappings is given by the universal property defining the direct product. Indeed, categorically the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152027.png" /> is an object together with morphisms: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152028.png" /> such that for every family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152029.png" /> there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152031.png" />.
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In an arbitrary category with products, cf. [[Direct product|Direct product]], the diagonal product of mappings is given by the universal property defining the direct product. Indeed, categorically the product $  Y = \prod _  \alpha  Y _  \alpha  $
 +
is an object together with morphisms: $  \pi _  \alpha  : Y \rightarrow Y _  \alpha  $
 +
such that for every family of morphisms $  \phi _  \alpha  : X \rightarrow Y _  \alpha  $
 +
there is a unique morphism $  f : X \rightarrow Y $
 +
such that $  \pi _  \alpha  f = f _  \alpha  $.
  
Tikhonov's imbedding result is in [[#References|[a2]]]. E. Čech, inspired by Tikhonov's result, obtained the following imbedding theorem [[#References|[a1]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152032.png" /> be the family of continuous mappings from a [[Completely-regular space|completely-regular space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152033.png" /> into the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152034.png" />. Then the diagonal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152035.png" /> is an imbedding, and the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152037.png" /> is equivalent to the [[Stone–Čech compactification|Stone–Čech compactification]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031520/d03152038.png" />.
+
Tikhonov's imbedding result is in [[#References|[a2]]]. E. Čech, inspired by Tikhonov's result, obtained the following imbedding theorem [[#References|[a1]]]: Let $  {\mathcal C} $
 +
be the family of continuous mappings from a [[Completely-regular space|completely-regular space]] $  X $
 +
into the unit interval $  I $.  
 +
Then the diagonal mapping $  F: X \rightarrow I ^  {\mathcal C}  $
 +
is an imbedding, and the closure of $  F ( X) $
 +
in $  I ^  {\mathcal C}  $
 +
is equivalent to the [[Stone–Čech compactification|Stone–Čech compactification]] of $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,  "On bicompact spaces"  ''Ann. of Math.'' , '''38'''  (1937)  pp. 823–844</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  "Ueber die topologische Erweiterung von Räumen"  ''Math. Ann.'' , '''102'''  (1929)  pp. 544–561</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,  "On bicompact spaces"  ''Ann. of Math.'' , '''38'''  (1937)  pp. 823–844</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  "Ueber die topologische Erweiterung von Räumen"  ''Math. Ann.'' , '''102'''  (1929)  pp. 544–561</TD></TR></table>

Revision as of 17:33, 5 June 2020


$ f _ \alpha : X \rightarrow Y _ \alpha $, $ \alpha \in {\mathcal A} $

The mapping $ f: X \rightarrow Y = \prod \{ {Y _ \alpha } : {\alpha \in {\mathcal A} } \} $ defined by the equation $ f ( x) = \{ f _ \alpha ( x) \} \in Y $. The diagonal product of mappings $ f _ \alpha $ satisfies, for any $ \alpha $, the relation $ f _ \alpha = \pi _ \alpha f $, where $ \pi _ \alpha $ denotes the projection of the product $ Y $ on the factor $ Y _ \alpha $. The diagonal product of continuous mappings is continuous. A family of mappings $ f _ \alpha : X \rightarrow Y _ \alpha $ of topological spaces is said to be partitioning if for any point $ x \in X $ and neighbourhood $ Ox $ of $ x $ there exist an index $ \alpha $ and an open subset $ U _ \alpha \subset Y _ \alpha $ such that $ x \in f _ \alpha ^ { - 1 } U _ \alpha \subset Ox $. If $ \{ f _ \alpha : X \rightarrow Y _ \alpha \} $ is a partitioning family of mappings and if $ f $ is the diagonal product of the mappings $ f _ \alpha $, then $ f $ is an imbedding of $ X $ into the product $ \prod Y _ \alpha $, i.e. $ f: X \rightarrow fX $ is a homeomorphism. The diagonal product of mappings was used by A.N. Tikhonov to imbed a completely-regular space of weight $ \tau $ in the cube $ I ^ { \tau } $.

Comments

Instead of calling a family of mappings partitioning, one says that it separates points and closed sets.

In an arbitrary category with products, cf. Direct product, the diagonal product of mappings is given by the universal property defining the direct product. Indeed, categorically the product $ Y = \prod _ \alpha Y _ \alpha $ is an object together with morphisms: $ \pi _ \alpha : Y \rightarrow Y _ \alpha $ such that for every family of morphisms $ \phi _ \alpha : X \rightarrow Y _ \alpha $ there is a unique morphism $ f : X \rightarrow Y $ such that $ \pi _ \alpha f = f _ \alpha $.

Tikhonov's imbedding result is in [a2]. E. Čech, inspired by Tikhonov's result, obtained the following imbedding theorem [a1]: Let $ {\mathcal C} $ be the family of continuous mappings from a completely-regular space $ X $ into the unit interval $ I $. Then the diagonal mapping $ F: X \rightarrow I ^ {\mathcal C} $ is an imbedding, and the closure of $ F ( X) $ in $ I ^ {\mathcal C} $ is equivalent to the Stone–Čech compactification of $ X $.

References

[a1] E. Čech, "On bicompact spaces" Ann. of Math. , 38 (1937) pp. 823–844
[a2] A.N. [A.N. Tikhonov] Tichonoff, "Ueber die topologische Erweiterung von Räumen" Math. Ann. , 102 (1929) pp. 544–561
How to Cite This Entry:
Diagonal product of mappings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_product_of_mappings&oldid=14452
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article