Real Algebraic and Analytic Geometry
Submission: 2006, January 23.
Given a topological space $X$, $\K(X)$ denotes the upper semi-lattice of its (Hausdorff) compactifications. Recent studies have asked when, for $\ax\in \K(X)$, the restriction homomorphism $\rho\colon C(\ax)\to C(X)$ is an epimorphism in the category of commutative rings. This article continues this study by examining the sub-semilattice, $\Ke(X)$, of those compactifications where $\rho$ is an epimorphism along with two of its subsets, and its complement $\Kne(X)$. The role of $\K_z(X)\sbq \K(X)$ of those $\ax$ where $X$ is $z$-embedded in $\ax$, is also examined. The cases where $X$ is a $P$-space and, more particularly, where $X$ is discrete, receive special attention.
Mathematics Subject Classification (2000): 18A20, 54C40, 54C45.
Keywords and Phrases: Rings of continuous functions, ring epimorphisms, compactifications.
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