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\(C\)-epic compactifications. (English) Zbl 0993.54024
Let \(K\) be a compactification of the Tikhonov space \(X\), and \(\rho_K\colon C(K)\to C(X)\) the restriction operator. In case \(\rho_K\) is an epimorphism in the category of Archimedean \(l\)-groups with unit, then \(K\) is called a \(C\)-epic compactification of \(X\), or \(X\) is \(C\)-epic in \(K\). In this paper the authors study the question when \(X\) is \(C\)-epic in some compactification \(K\). They prove, for example, that if \(X\) is \(C\)-epic in \(K\) then \(K\) and \(\beta X\) have the same basically disconnected cover, and that \(\beta X\) is the only \(C\)-epic compactification of \(X\) if and only if \(X\) is pseudocompact. In addition, \(X\) is \(C\)-epic in each of its compactifications if and only if \(X\) is weakly Lindelöf. The paper contains many examples and there are several interesting open questions.

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
46A40 Ordered topological linear spaces, vector lattices
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54C45 \(C\)- and \(C^*\)-embedding
46J10 Banach algebras of continuous functions, function algebras
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
54C30 Real-valued functions in general topology
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
Full Text: DOI
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