On some properties of the metric dimension
Ladislav Mišík, Tibor Žáčik · Czech digital mathematics library · 1990
In the paper two covering functions JV, M defined on a given compact metric space K are studied; their binary logarithms are usually called e-entropy and e-capacity of this space, respectively.For a function u with suitable properties a compact countable metric space, for which the function u is the covering function, is constructed.By means of covering functions the both lower dim and upper dim metric dimensions of K are defined.It is shown that for a given compact metric space K and every a € [0, dim K] and fi 6 [O.dim K] there is a compact countable subspace X of K with the unique cluster point such that dim X = a and dim X < /?.Finally, it is shown that there exist compact spaces with arbitrary small dim which are not isometrically embeddable into R m for each m € N.