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

114 S. Ceretková, J. Fulier and J. T. Tóth Then /(<*') >/(d), so d! # f~

l({t}). (ii) According to lemma the set <71(0}) i

s closed in B. It is enough to show that the set is dense in B. Let d E flf1({i}) 311(

1 0 < £ < 1. Define d' on X as follows: f

]'

(r v\ _íd{x,y)+ f, ifx^y if x = y Evidently g(d') = t + f

, therefore g' £ B \ and d*(d,d') < e. (iii) We can prove similarly hke (ii). • From the above Theorem 1 we can see that the sets /

_

1 ({*})> h~

l({t}) are small from the topological point of view but on the other hand we show, that the cardinality of them is equal to the cardinality of the set M. In [2] is proved: card(Aí) = c if X is a finite set having at least two elements and card (M) = 2

car

d^

A^ if X is infinite set (c denotes the cardinality of the set of all real numbers). Theorem 2. Let X be an infinite set. Then we have: 1. card(/1 ({*})) = 2

car

dW fort E (0,1] 2. card(<71({i})) = 2

car

dW for t E [0, -foo) • 3. card(/i

_

1 ({t})) = 2

car

dW for t E (0, +oo). Proof. 1. Let 0 < t < 1 and 0 < £ < \ • . Let B C X for which card(P) > 2. We define the metric on X as follows: {0, if x = y iix,yeB;x^y - £, if x £ B or y B, x ± y It is to easy to verify that <jß is a metric and that o

ß / v

'ß, if B B'

. Evidently /(

Gß

) = t. There are 2

car

d(

x) many choices for B so we can see 2card(X) < car

d(/-i ({*})) < card(A^) < 2

card(x

). We get by the Cantor-Bernstein theorem that card(/

_

1 ({/})) = 2

car

dW. Let now t — 1 and XQ = {xi < x

2 < • • • < x

n < • • •} C X

. Define the function dß'-X X X —• R: dß(x

n, x

m) = \n — m\ for n,m=l,2,

... d

B(x,x

n) = d

B(x

n,x) = n for x £ X

0 dß{x,y) = d

B(y,x) = 1 for x,y^X

0,x^y dß{x,x) = 0 for x E X.