View source for Diagonal product of mappings

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'' $  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|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  $  {\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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,   "On bicompact spaces"  ''Ann. of Math.'' , '''38'''  (1937)  pp. 823–844 {{ZBL|63.0570.02}}</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>