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 113 Then we required inequality follows from (1) and (2). Analogously |h(di) — h(d 2)\ < d*(di,d 2). • Remark 1. The function h can be continuosly continued on A4. Because the set B is closed in A4 , the Hausdorff's function (see [1], p. 382) is continuous continuation of the function h on A4. Remark 2. Because AU B = M, AC\ B = $ and the set B is closed in M, according the lemma of Uryshon there exists a function G\ A4 [0,1] such that G is continuous on A4 and G(A) = {0}, G(B) = {1}. For this reason G(M) = {0,1}. Space (A4,d*) is Bair's space, e.g. every non-empty open subset of the set A4 is of the 2-nd cathegory in A4. The set A is non-void and open subset in M, then the set / _ 1({1}) = A is of the 2-nd cathegory in A4. One may ask: Is there any t E (0,1) such that the set / _ 1({t}) is of the 2-nd cathegory in A4? Similarly for 5 _ 1({i}) and This question is answered in the next theorem. Theorem 1. We have (i) For arbitratry t E (0,1) the set f~ l({t}) is nowhere dense in A4. (ii) For arbitrary t E [0, +00) the set g~~ l({t}) is nowhere dense in A4. (iii) For arbitrary t E [0, +00) the set /i1({t}) is nowhere dense in A4. Proof, (i) Let 0 < t < 1. According to lemma the set /1({i}) is closed in A4. Therefore it is sufficient to prove that the set M \ /1(0)) is dense in A4. We will use inequality ^2 11 12 — /1 (3) —— > —— + for 0 < h < t 2 K ; l + t 2 - 1-Mi (l + t 2) 2 - (it is equivalent to (i 2 — t\ ) 2 >0). Let d E f~ l{{t}) and 0 < £ < 1. Clearly d E B and there exists a K E R+ such that (4) d(x,y) < K for every x,y E X. Choose d' E A4 as follows _Jd(x,y) + §, ifx,yEx,x/y d'(x,y ) 2 0, if x = y. Then d*(d,d') < e. We show that d' E A4 \ / _ 1 ({*})• ( 3) and (4) for x, y E X(x / y) and — d(x, y), t 2 = d'(x, y) we have £ e <p(d'(x,y)) > tp(d(x,y)) + 77—7^7 TTT > <p{d{x,y)) + (1 + d'(x,y)) 2 v ' (1 + K) 2 '
