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Difference between revisions of "Fibre product"

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The  
 
The  
''fibre product'' of a system of topological spaces $\def\a{\alpha}X_\a$
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''fibre product'' of a system of topological spaces $\def\a{\alpha}X_\a$ with respect to a system of continuous mappings $f_\a:X_\a\to X_0$,
with respect to a system of continuous mappings $f_\a:X_\a\to X_0$,
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$\a\in\def\cA{ {\mathcal A}}\cA$ is the subset $X_\cA$ of the [[Tikhonov product]] $\prod_{\a\in\cA}X_\a$, which is considered in the [[induced topology]] and which consists of the points
$\a\in\def\cA{ {\mathcal A}}\cA$ is
 
the subset $X_\cA$ of the
 
[[Tikhonov product|Tikhonov product]] $\prod_{\a\in\cA}X_\a$, which is
 
considered in the induced topology and which consists of the points
 
 
$x=\{x_\a\}\in \prod_{\a\in\cA}X_\a$ for which $f_\a x_\a = f_{\a'}
 
$x=\{x_\a\}\in \prod_{\a\in\cA}X_\a$ for which $f_\a x_\a = f_{\a'}
 
x_{\a'}$, for all indices $\a$ and $\a'$ from $\cA$. The mapping which
 
x_{\a'}$, for all indices $\a$ and $\a'$ from $\cA$. The mapping which
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====Comments====
 
====Comments====
In category theory the term  "pullback"  is also used, cf.
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In category theory the term  "pullback"  is also used, cf. [[Fibre product of objects in a category]].
[[Fibre product of objects in a category|Fibre product of objects in a category]].
 

Latest revision as of 15:27, 20 November 2014

2020 Mathematics Subject Classification: Primary: 54B [MSN][ZBL]


The fibre product of a system of topological spaces $\def\a{\alpha}X_\a$ with respect to a system of continuous mappings $f_\a:X_\a\to X_0$, $\a\in\def\cA{ {\mathcal A}}\cA$ is the subset $X_\cA$ of the Tikhonov product $\prod_{\a\in\cA}X_\a$, which is considered in the induced topology and which consists of the points $x=\{x_\a\}\in \prod_{\a\in\cA}X_\a$ for which $f_\a x_\a = f_{\a'} x_{\a'}$, for all indices $\a$ and $\a'$ from $\cA$. The mapping which brings the point $x=\{x_\a\}\in X_\cA$ into correspondence with the point $x_\a\in X_\a$ (or with the point $f_\a x_\a\in X_0$) is called a projection of the fibre product $X_\cA$ onto $X_\a$, $\a\in\cA$ (or onto $X_0$). If the space $X_0$ is a one-point space, then $X_\cA\cong \prod_{\a\in\cA}X_\a$. If the $X_\a$, $\a\in\cA$, are completely-regular spaces, the fibre product $X_\cA$ is completely regular. The fibre product, in particular its special case the partial product, is well suited for the construction of universal (in the sense of homeomorphic inclusion) topological spaces of given weight and given dimension (cf. Universal space).


Comments

In category theory the term "pullback" is also used, cf. Fibre product of objects in a category.

How to Cite This Entry:
Fibre product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fibre_product&oldid=30736
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article