# Fibre product

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).