On generating heavy points with positive matrices

Abstract

Let S be a countable topological space. If S is indexed by the positive integers, then real functions on S may be considered as sequences and are subject to infinite matrix transformation. After Henriksen and Isbell in [3], a point s in S is called a heavy point if there exists a regular matrix which sums every bounded real-valued function on S continuous at s. It is clear that the notion of a heavy point is independent of the order chosen for S. Some properties of heavy points are given in [3 ]. For example, sequential limit points are heavy points and isolated points (in fact, points whose complements are C*embedded) are not heavy points. In addition Henriksen and Isbell observe that not all heavy points are sequential limit points. The latter phenomenon was the motivation for the present paper. Corresponding to each positive regular matrix A we define a topology on the positive integers by taking neighborhoods of 1 to be sets whose characteristic functions are A-summable to 1. Denote the resulting space by NA. The point 1 is a heavy point in NA. Some basic properties of these spaces are examined. Also, we determine the class of matrices A for which 1 is not a sequential limit point in NA. Let A = (ank) be an infinite real matrix. The A-transform of the real sequence x = { xk } is the sequence { Sk ankXk } if these sums exist. Let c and m denote the Banach spaces of convergent sequences and bounded sequences, respectively, with jjx|I =supk| X . Let CA = {x: AxEc}. The matrix A is said to sum the members of CA or any subset of CA. Define the functional limA on CA by limAx=lim Ax. The matrix A is called conservative if CCCA and regular if in addition lim =limA on c. It is well known that A is regular if and only if limn ank=O for all k, limn Ekank=l1 and SUpn Zk lank! < o. The matrix A is called positive if ank _ 0 for all n, k. For a set T of positive integers let x(T) denote the characteristic function of T; (X(T))k =1 for k E T, 0 otherwise. For a topological space S let C*(S) denote the Banach space of bounded real-valued continuous functions on S with ||x|| =sup { I x(s)|: sES}. As observed in [3], if S is countable and completely regular, then S is 0-dimensional (i.e. the ring r of open and