Fibre product

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2010 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).


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:
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article