Three Cardinal Functions Similar to Net Weight

The purpose of this paper is to introduce and investigate cardinal functions called pseudonet weight, weak net weight, and weak pseudonet weight. These are similar to but generally smaller than net weight. We look at how these cardinal functions relate to hereditary Lindelöf degree, hereditary density, and spread, and we study their behavior under products. An important and useful cardinal function for a topological space is that of weight, namely, the minimum cardinal for a base of open sets. Net weight is similar to weight, except that "base" members need not be open. In this paper we look at three cardinal functions which are slight variations of net weight. Throughout this paper, k denotes an infinite cardinal number, and for simplicity, all cardinal functions will be infinite. The smallest (infinite) cardinal number k such that X is hereditarily /c-Lindelöf (hereditarily «r-separable, resp.) is denoted by hl(X) ihdiX), resp.). The spread of X (equivalently, hereditary Souslin number) is denoted by siX). As usual, wiX) denotes the weight of X. The set of all real numbers is denoted by R . For notation and terminology not defined here, see [1]. In § 1 we generalize the notion of nets (also called networks) by introducing «r-pseudonets. Some examples are given of (nonregular) Hausdorff spaces which have K-pseudonets but have no nets of cardinality < k . Theorem 1.9 shows that pseudonet weight coincides with net weight in regular spaces. We also examine weak net weight and weak pseudonet weight. In terms of definition, weak net weight is to net weight as weak pseudonet weight is to pseudonet weight. The main theorem in §2 is Theorem 2.3, which shows that the Cartesian product X x Y of a hereditarily K-Lindelof space X and a space F having a k-pseudonet is hereditarily «r-Lindelof. 1. Definitions, examples, and elementary relationships 1.1. Definition (cf. [1, Remark 3.1.17, p. 170]). A family f of subsets of a topological space X is called a net in X if and only if for each open set Received by the editors March 2, 1989 and, in revised form, July 21, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 54A25.