# zbMATH — the first resource for mathematics

$$C$$-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
##### Keywords:
compactification; lattice-ordered group; $$f$$-ring
Full Text: