Cardinal functions for 𝑘-spaces

Abstract

Abstract. In this paper, four cardinal functions are defined on the class of fc-spaces. Some of the relationships between these cardinal functions are studied. Characterizations of various ¿-spaces are presented in terms of the existence of these cardinal functions. A bound for the ordinal invariant k of Arhangelskii and Franklin is established in terms of the tightness of the space. Examples are presented which exhibit the interaction between these cardinal invariants and the ordinal invariants of Arhangelskii and Franklin.1. Introduction. The class of ¿-spaces, introduced by Arens [1], has become a widely used class of topological spaces, both because of their intrinsic interest and because they provide a natural category of spaces in which to study a variety of other topological problems. The ¿-spaces are precisely those spaces which are the quotient spaces of locally compact spaces. In particular, X is a k-space provided: H is closed in X if and only if H n K is closed in K, for each compact set K. That is, X has the weak topology induced by the collection of compact subspaces. Franklin [5] initiated the study of a restricted class of ¿-spaces, called sequential spaces. This is the class of ¿-spaces in which the weak topology is induced by the collection of convergent sequences. If A c X, let k c\(A) be the set of all points/? such that p E c\K(A n K) for some compact set K, and let s cl(A) be the set of all points/» such that there exists a sequence in A that converges top. Arhangelskii and Franklin [3] introduced the concept of ordinal invariants for topological spaces, which will be called here the compact order and the sequential order of a space. Let A be a subset of a topological space X. Let A0 = A, and for each nonlimit ordinal a = ß + l,\etAa = k c\(Aß) [Aa = s cl(A ß)]. If a is a limit ordinal, let A " = U {A ß: ß < a). The compact order [sequential order] of X is defined as

A note on linearly ordered net spaces

“…Consider the space S 2 = {(0, 0)} U {(1/n, 0):neN}{J {Q/n, 1/m): n, m € N} in [1] and [4]. Let U n = {(1/n, 1/m): m e N}.…”

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)).…”

“…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.…”