上盒维数和填充维数

 

Let $X$ be a totally bounded metric space. 

(1) If $X$ is compact and if $\overline{\dim}_MU\ge s$ for every non-empty open set $U\subset X,$ then $\dim_PX\ge s.$

(2)  If $\dim_PX>s,$ then there is a closed set $C\subset X$ such that $\dim_P(C \cap U)>s$ for every open set $U$ which intersects $C.$

For part (1) see Falconer, Fractal Geometry, Proposition 3.9, for part (2) see Falconer and Howroyd, Projection theorems for box and packing dimensions, Lemma4.