z-congruences and topologies on $C^+(X)$

Abstract

For a Tychonoff space X, $C^+(X)$ denotes the non-negative real-valued continuous functions on X. We obtain a correlation between z-congruences on the ring C(X) and z-congruences on the semiring $C^+(X)$. We give a new characterization of P-spaces via z-congruences on $C^+(X)$. The z-congruences on $C^+(X)$ are shown to have an algebraic nature like z-ideals. We study some topological properties of $C^+(X)$ under u-topology and m-topology. It is shown that a proper ideal of $C^+(X)$ is closed under m-topology if and only if it is the intersection of maximal ideals of $C^+(X)$. Also, we prove that every ideal of $C^+(X)$ is closed if and only if X is a P-space. We investigate the connectedness and compactness of $C^+(X)$ under m-topology. It is shown that the component of $\varvec{0}$ is $C_\psi (X)\cap C^+(X)$. Finally, we show that $C_m^+(X)$ is locally compact, $\sigma$-compact and hemicompact if and only if X is finite.