Reflection Theorems for Cardinal Functions
by
Richard E. Hodel
Duke University
Coauthors: Jerry Vaughan (UNCG)

Let f be a cardinal function and let k be an infinite cardinal. We say that f reflects k if the following holds: if f(X) is greater than or equal to k, then there is a subspace Y of X such that f(Y) is greater than or equal to k and the cardinality of Y is at most k+ (successor of k). Example: for f = weight and k = w we have: if X has an no countable base, then there is a subspace of X of cardinality at most w+ that has no countable base (due to Hajnal-Juhasz). In this joint work with J. Vaughan, we study a number of properties related to reflection: strong reflection, the Darboux property, and the increasing union property. The following cardinal functions are considered: cellularity, extent, Lindelof degree, density, hereditary density, hereditary Lindelof degree, net weight, pi-weight, character, tightness, spread, point-weight.

Date received: June 5, 1999