\(C^*\)-embedding, and \(C\)-embedding on product spaces
A subspace \(A\) of a space \(X\) is said to be \(C\)-embedded (resp. \(C^\ast\)-embedded) in \(X\) if every real-valued, continuous (resp. bounded continuous) function on \(A\) can be extended over \(X\). An interesting problem in extension theory is under what conditions on \(A\) and \(X\) a \(C^\ast\)-embedded closed subspace \(A\) of \(X\) is \(C\)-embedded in \(X\). In particular, the following problem is open: Assume that \(A\) is a \(C\)-embedded closed subspace of a Tychonoff space \(X\), \(Y\) is a space and \(A\times Y\) is \(C^\ast\)-embedded in \(X\times Y\). Then, is \(A\times Y\) \(C\)-embedded in \(X\times Y\)? The author proves that the answer to the problem above is positive if \(Y\) is a \(\sigma\)-locally compact, paracompact, Hausdorff space. Moreover, she proves: (1) If \(A\) is a \(C\)-embedded subspace of a normal \(P\)-space \(X\), \(Y\) is a paracompact, Hausdorff \(\Sigma\)-space and \(A\times Y\) is \(C^\ast\)-embedded in \(X\times Y\), then \(A\times Y\) is \(C\)-embedded in \(X\times Y\); and (2) if \(A\) is a \(P\)-embedded subspace of a normal \(P\)-space \(X\), \(Y\) is a paracompact, Hausdorff \(\sigma\)-space and \(A\times Y\) is \(C^\ast\)-embedded in \(X\times Y\), then \(A\times Y\) is \(P\)-embedded in \(X\times Y\).
scientific article; zbMATH DE number 1102904