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Ĉech-Stone-like compactifications for general topological spaces. (English) Zbl 0754.54014

The author considers the well-known problem whether every topological space \(X\) has a compactification \(Y\) such that every continuous mapping \(f\) from \(X\) into a compact space \(Z\) has a continuous extension from \(Y\) in \(Z\). In categorical terms it can be formulated as follows: whether every topological space has a reflection (weak reflection) in the class of compact spaces. In general, the answer is negative but for some spaces such compactifications (reflections) exist. In the paper two theorems and some corollaries are proved. Theorem 1: If the Wallman remainder of \(X\) is finite, then the Wallman compactification of \(X\) is the weak reflection of \(X\) in compact spaces. – Theorem 2: If \(X\) contains an infinite family \(\{X_ n\}\) of closed noncompact subsets such that \(X_ n\cap X_ m\) is compact for \(n\neq m\), then \(X\) has no weak reflection in compact spaces.

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54C20 Extension of maps
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54B30 Categorical methods in general topology
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