The $lambda$-super socle of the ring of continuous functions

Abstract

The concept of $lambda$-super socle of $C(X)$, denoted by $S_lambda(X)$ (i.e., the set of elements of $C(X)$ such that the cardinality of their cozerosets are less than $lambda$, where $lambda$ is a regular cardinal number with $lambdaleq |X|$) is introduced and studied. Using this concept we extend some of the basic results concerning $SC_F(X)$, the super socle of $C(X)$ to $S_lambda(X)$, where $lambda geqaleph_0$. In particular, we determine spaces $X$ for which $SC_F(X)$ and $S_lambda(X)$ coincide. The one-point $lambda$-compactification of a discrete space is algebraically characterized via the concept of $lambda$-super socle. In fact we show that $X$ is the one-point $lambda$-compactification of a discrete space $Y$ if and only if $S_lambda(X)$ is a regular ideal and $S_lambda(X)=O_x$, for some $xin X$.

Keywords

• $lambda$-super socle
• $lambda$-isolated point
• one point $lambda$-compactification
• $p_lambda$-space