Previous |  Up |  Next


metric dimension; covering dimension; $\varepsilon $-translation; uniformly $0$-dimensional mappings
Some theorems characterizing the metric and covering dimension of arbitrary subspaces in a Euclidean space will be obtained in terms of $\varepsilon $-translations; some of them were proved in our previous paper [G1] under the additional assumption of the boundedness of subspaces.
[AH] Alexandroff P., Hopf H.: Topologie. Berlin, Springer-Verlag, 1935. Zbl 0277.55001
[Eg] Egorov V.I.: On the metric dimension of points of sets (in Russian). Mat. Sb. 48 (1959), 227-250. MR 0125563
[E] Engelking R.: Dimension Theory. North Holland, 1978. MR 0482697 | Zbl 0401.54029
[G1] Goto T.: Metric dimension of bounded subspaces in Euclidean spaces. Top. Proc. 16 (1991), 45-51. MR 1206452
[G2] Goto T.: A construction of a subspace in Euclidean space with designated values of dimension and metric dimension. Proc. Amer. Math. Soc. 118 (1993), 1319-1321. MR 1189543 | Zbl 0805.55002
[Ka1] Katětov M.: On the dimension of non-separable spaces I (in Russian). Czech. Math. J. 2 (1952), 333-368. MR 0061372
[Ka2] Katětov M.: On the relation between the metric and topological dimensions (in Russian). Czech. Math. J. 8 (1958), 163-166. MR 0105084
[Ku] Kuratowski K.: Topology I. New York, 1966. Zbl 0849.01044
[S] Sitnikov K.: An example of a two dimensional set in three dimensional Euclidean space allowing arbitrarily small deformations into a one dimensional polyhedron and a certain new characterization of the dimension of sets in Euclidean spaces (in Russian). Dokl. Akad. Nauk SSSR 88 (1953), 21-24. MR 0054245
[Sm1] Smirnov Ju.: On the metric dimension in the sense of P.S. Alexandroff (in Russian). Izv. Akad. Nauk SSSR 20 (1956), 679-684. MR 0082096
[Sm2] Smirnov Ju.: Geometry of infinite uniform complexes and $\delta $-dimension of points sets. Mat. Sb. 38 (1956), 137-156; Amer. Math. Soc. Transl. Ser. 2, 15 (1960), 95-113. MR 0115158
[Z-S] Zarelua A., Smirnov Ju.: Essential and zero-dimensional mappings. Dokl. Nauk SSSR 148 (1963), 1017-1019. MR 0157351 | Zbl 0129.38501
Partner of
EuDML logo