CLASSES OF LOCALLY FINITE COLLECTIONS IN TOPOLOGICAL AND UNIFORM SPACES

In this paper we study several classes of locally finite collections. We show their interrelationships and how they relate to known topological concepts such as para-compact and expandable. If (X,J) is a topological space and if n E ~, we say that a cover § of X is n-even if there exist neighborhoods wl,···,w of the diagonal of X such that W ~ c W. 1 for n 1 1 i = 2,···,n and (Wl(x))xEX refines §. (Here W(x) = {y E X: (x,y) E W}.) If (X,U) is a uniform space we say that a cover § of X is Lebesgue if there exists a U in U such that (U(x))xEX refines §. If every open cover of X is Lebesgue we say that X has the Lebesgue property. We write Uo for the universal uniformity on X: i.e