Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

CERETKOVÁ, S., FULIER, J. and TÓTH J. T., On the certain subsets of the space of metrics

On the certain subsets of the space of metrics S. CERETKOVÁ, J. FULIER and J. T. TÓTH Abstract. In this note we look at certain subsets of the metric space of metrics for an arbitrary given set X and show that in terms of cardinility these can be very large while being extremely small in the topological point of view. Let X be a given non-void set. Denote by Ai the set of all metrics on X endowed with the metric: d*(di

, d

2) = minjl, sup {\di(x, y) — d

2(x, ?/)[} for di

, d

2 £ M }. Results shown in [2] include A4 is a non-complete Baire space and 7i is an open and dense subset of A4, thus M \ H is nowhere dense in A4. Other results on the metric space of metrics may be found in [2], [3] and [4]. Let A and B denote the set of all metrics on X that are unbounded and bounded, respectively. It is proved in [2] (Theorem 5) that A,B are non-empty, open subsets of the Baire space (M.,d*) (of [2], Theorem 3) provided X is infinite. Thereby A,B are sets of the 2-nd cathegory in A4, if is infinite. If X is finite, then B = A4 and .4 = 0. Now define the mapping Introduction x,y EX First of all recall some basic definitions and notations. Suppose a > 0 and put 7i

a = {de M : V d(x,y) > a} and H = [ J H a • f: A4 (0,1], g:M [0,oo) and h:B (0, +oo) as follows: where d £ A4, g{d

) = inf d(x,y) where d £ M, and h(d) = sup d(x,y) where d £ B. x,y£X