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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.

MSC:
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
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