Abstract

Intermediate rings A(X, K) of K-valued continuous functions lying between B(X, K) and C(X, K), defined over a zero dimensional space X, are investigated and studied in this article, here K stands for a countable subfield of ℝ. It is realized that the structure space of A(X, K) is β₀X, the Banaschewski compactification of X. The Hewitt realcompactification analogue υ_K (X) of υX is defined. Several equivalent descriptions of pseudocompactness of X via υ_K (X), β₀X and the uniform topology and the m-topology on C(X, K) are given. The article ends after studying an intercorrelation between z-ideals and z°-ideals in the rings C(X, K) and C_c(X), the functionally countable subalgebra of C(X). This study eventually leads to a characterization of P -spaces and almost P-spaces in terms of z-ideals and z°-ideals in C(X, K).

摘要

本文研究和研究了位于B ( X, K ) 和C ( X, K )之间的K值连续函数的中间环A ( X, K )，定义在零维空间X上，这里K代表ℝ 的可数子域。可知A ( X, K )的结构空间为β ₀ X ，即X的 Banaschewski 紧化。υX的 Hewitt 实紧缩模拟υ _K ( X )被定义为。给出了通过υ _K ( X )、β ₀ X以及C ( X, K )上的均匀拓扑和m拓扑对X的赝紧性的若干等价描述。本文在研究环C ( X, K ) 和C _c ( X ) 中的z理想和z °理想之间的相互关系后结束， C ( X ) 的功能可数子代数。这项研究最终导致了P的表征-spaces 和几乎P -spaces 根据C ( X, K )中的z理想和z °理想。