Compactifications, C(X) and ring epimorphisms

W.D. Burgess and R. Raphael

Given a topological space $X$, $K(X)$ denotes the upper semi-lattice of its (Hausdorff) compactifications. Recent studies have asked when, for $\alpha X \in K(X)$, the restriction homomorphism $\rho : C(\alpha X) \to C(X)$ is an epimorphism in the category of commutative rings. This article continues this study by examining the sub-semilattice, $K_{epi}(X)$, of those compactifications where $\rho$ is an epimorphism along with two of its subsets, and its complement $K_{nepi}(X)$. The role of $K_z(X)\subseteq K(X)$ of those $\alpha X$ where $X$ is $z$-embedded in $\alpha X$, is also examined. The cases where $X$ is a $P$-space and, more particularly, where $X$ is discrete, receive special attention.

Keywords: epimorphism, ring of continuous functions, category of rings, compactifications

2000 MSC: 18A20, 54C45, 54B40

Theory and Applications of Categories, Vol. 16, 2006, No. 21, pp 558-584.

TAC Home