Cardinal Functions for k-Spaces

On open mappings, closed mappings, and ordinal invariants

“…For each ordinal a the ath k-closure is A = A and A = k cl(A) » if a = B + 1 , then A a = k cl [A ) , and if a is a limit ordinal For a complete discussion of the properties of compact order see [J], and for a presentation of cardinal invariants see [3]. Recently in [2]…”

Universal bounds and monotonicity of ordinal invariants

“…Introduction. The bounds for the ordinal invariants sequential order a of sequential spaces and compact order k of /t-spaces have been determined as o(X) < u3x [2] and k(X) < t(X)+ [7] where t(X)+ is the successor of the tightness of X. These ordinal invariants are monotonie decreasing for pseudo-open mappings [3] in the sense that, if A is a A>space (sequential space) and /: A* -» Y is a continuous pseudo-open surjection then <c(A") > k(Y) (o(X) > a(Y)).…”

“…Proof. Since the quotient space of a 2-space is a 2-space [8], y is a 2-space and 2 (7) exists. Suppose 2(A") = p.. Let B be any subset of Y.…”

“…The various theorems concerning the cardinal invariants in sequential and A>spaces in [7] are also true in the general setting of natural covers where we would consider the 2-cardinal of A and the pointwise 2-cardinal of X.…”