Products of compactifications
Let \(X\) be a completely regular (Hausdorff) topological space and \(b_1X\), \(b_2X\) two compactifications of \(X\). The natural inclusions \(i_k\:X \to b_kX\), \(k = 1,2\) define a new inclusion \(i\:X \to b_1X \times b_2X\) by \(i(x) = (i_1(x), i_2(x))\) for \(x\in X\). The closure of \(i(X)\) in \(b_1X \times b_2X\) is some new compactification \((b_1\times b_2)X\) of \(X\). The authors call it the product of the compactifications \(b_1X\) and \(b_2X\) of the space \(X\). They study the minimality of the compactification in the family of all compactifications of \(X\) following the previous ones (with respect to an order relation \(\leq\) by continuous surjections preserving \(X\)), the associativity property and several properties. The product \((\prod_\lambda b_\lambda)X\) of finite or infinite compactifications of \(X\) can also be considered in a similar way. The authors show that \(\varprojlim(b_{\lambda_1}\times\dots\times b_{\lambda_k})X = (\prod_\lambda b_\lambda)X\), here \(\{b_\lambda X\}\) is an infinite collection of compactifications of \(X\) and the inverse system \(\{(b_{\lambda_1}\times\dots\times b_{\lambda_k})X\}\) is naturally defined by the collection. Then an application of the theorem to a special compactification of a locally compact space is considered.
scientific article; zbMATH DE number 1870470