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Difference between revisions of "Fibre product"

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''of a system of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400401.png" /> with respect to a system of continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400403.png" />''
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The subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400404.png" /> of the [[Tikhonov product|Tikhonov product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400405.png" />, which is considered in the induced topology and which consists of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400406.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400407.png" />, for all indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f0400409.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004010.png" />. The mapping which brings the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004011.png" /> into correspondence with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004012.png" /> (or with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004013.png" />) is called a projection of the fibre product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004014.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004016.png" /> (or onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004017.png" />). If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004018.png" /> is a one-point space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004019.png" />. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004021.png" />, are completely-regular spaces, the fibre product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040040/f04004022.png" /> 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|Universal space]]).
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The  
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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 $f_\a:X_\a\to X_0$,
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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
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$x=\{x_\a\}\in \prod_{\a\in\cA}X_\a$ for which $f_\a x_\a = f_{\a'}
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x_{\a'}$, for all indices $\a$ and $\a'$ from $\cA$. The mapping which
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brings the point $x=\{x_\a\}\in X_\cA$ into correspondence with the
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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.
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[[Universal space|Universal space]]).
  
  
  
 
====Comments====
 
====Comments====
In category theory the term  "pullbackpullback"  is also used, cf. [[Fibre product of objects in a category|Fibre product of objects in a category]].
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In category theory the term  "pullback"  is also used, cf. [[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=17578
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article