It is shown that every Polish space X with admits a compact subspace Y such that where and denote the topological and Hausdorff dimensions, respectively.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-3-2, author = {Tomasz Szarek and Maciej \'Sl\k eczka}, title = {On the Hausdorff Dimension of Topological Subspaces}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {54}, year = {2006}, pages = {201-205}, zbl = {1108.54020}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-3-2} }
Tomasz Szarek; Maciej Ślęczka. On the Hausdorff Dimension of Topological Subspaces. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) pp. 201-205. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-3-2/