Let $(X,d)$ be a metric space having Hausdorff dimension $\alpha>0$ and let $0<\beta<\alpha$. Is there a metric subspace of $X$ having Hausdorff dimension $\beta$?
By Corollary 7 in [How95], every analytic subset of a complete separable metric space which has positive (or infinite) Hausdorff measure of dimension s contains a compact set which has finite and positive Hausdorff measure of dimension s.
[How95] J.D. Howroyd. On dimension and on the existence of sets of finite positive Hausdorff measure. Proc. London Math. Soc. (3), 70:581–604, 1995.