### A Generalization of the Strict Topology.

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For X a Tikhonov space, let F(X) be the algebra of all real-valued continuous functions on X that assume only finitely many values outside some compact subset. We show that F(X) generates a compactification γX of X if and only if X has a base of open sets whose boundaries have compact neighborhoods, and we note that if this happens then γX is the Freudenthal compactification of X. For X Hausdorff and locally compact, we establish an isomorphism between the lattice of all subalgebras of $F\left(X\right)/{C}_{K}\left(X\right)$ and the...

Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S) → C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched) covering, i.e., an open and closed map whose fibres are finite, if and only if the induced homomorphism...

This paper is devoted to the study of algebraic properties of rings of continuous functions. Our aim is to show that these rings, even if they are highly non-noetherian, have properties quite similar to the elementary properties of noetherian rings: we give going-up and going-down theorems, a characterization of z-ideals and of primary ideals having as radical a maximal ideal and a flatness criterion which is entirely analogous to the one for modules over principal ideal domains.

Relations between homomorphisms on a real function algebra and different properties (such as being inverse-closed and closed under bounded inversion) are studied.

As usual $C\left(X\right)$ will denote the ring of real-valued continuous functions on a Tychonoff space $X$. It is well-known that if $X$ and $Y$ are realcompact spaces such that $C\left(X\right)$ and $C\left(Y\right)$ are isomorphic, then $X$ and $Y$ are homeomorphic; that is $C\left(X\right)$determines$X$. The restriction to realcompact spaces stems from the fact that $C\left(X\right)$ and $C\left(\upsilon X\right)$ are isomorphic, where $\upsilon X$ is the (Hewitt) realcompactification of $X$. In this note, a class of locally compact spaces $X$ that includes properly the class of locally compact realcompact spaces is exhibited...