Vol. 4 No 9 - Pi Mu Epsilon

AN EASIER CONDITIONTHAN TOTAL BOUNDEDNESSDaniel E. PutnamUniversity of IllinoisThe condition of total boundedness is a useful one: for instance,in establishing compactness. Therefore it is worthwhile to findsimpler properties equivalent to total boundedness. The conditionI suggest is this:Definition: A subset A of a metric space (X,d) is Cauchy bounded,if for all E > 0 and all infinite subsets BCA there are two pointsx,y E B with d(x,y)O there is an E nef consisting of a finite subset,{a.,a,...a 1, of A so that for any x e A we have d(x,ai)0, we could find x,y e B with d(x.y)