\(C\)-epic compactifications (Q5957288)

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.