on the sλ(x) and λ-zero dimensional spaces

Let sλ(x) = f c(x) : x z(f)< λ, such that λ is a regular cardinal number with λ x. it is generalization of cf(x) = sℵ0 (x) and scf(x) = sℵ1 (x). using this concept we extend some of the basic results concerning the socle to sλ(x). it is shown that if x is a λ-pseudo discrete space, then ck,λ(x) sλ(x). sλ-completely regular spaces are investigated. consequently, x is a sℵ1-completely regular space if and only if x is 1-zero dimensional space. sλp-spaces are introduced and studied.