On \(a\)-Kasch spaces. (English) Zbl 1240.54064

Estaji, Ali Akbar; Henriksen, Melvin

Arch. Math., Brno 46, No. 4, 251-262 (2010).

For an infinite cardinal \(a\), a Tychonoff space is called \(a\)-Kasch if its ring of real-valued continuous functions does not contain any essential ideal (i.e., ideal whose intersection with every non-zero ideal is non-trivial) with a generating set of cardinality less than \(a\). In this paper, basic properties of \(a\)-Kasch spaces are studied. In particular, it is shown that \(\aleph _0\)-Kasch spaces are precisely almost \(P\)-spaces (i.e., every non-empty \(G_{\delta }\)-set has a non-empty interior) and that \(\aleph _1\)-Kasch spaces are precisely pseudocompact and almost \(P\)-spaces.

Reviewer: Michal Kunc (Brno)

MSC:

54C40	Algebraic properties of function spaces in general topology
13A30	Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
54G10	\(P\)-spaces

Keywords:

\(a\)-Kasch space; almost \(P\)-space; \(C\)-embedded; essential ideal; extremally disconnected; fixed ideal; free ideal; Kasch ring; \(P\)-space