Dimension and structure of typical compact sets, continua and curves (Q1117053)

In the sense of Baire categories most compact subsets of a complete metric space \(<X,\rho >\) have Hausdorff and lower entropy dimension 0. If the compact sets having lower entropy dimension \(\geq \delta (>0)\) are dense then most compact sets have upper entropy dimension \(\geq \delta\). Most compact sets C in X have the following property: For any \(x\in C\) and \(0<\epsilon \leq 1\) there are arbitrarily small \(\sigma >0\) such that the ``annulus'' \(\{y\in X:\quad \epsilon \sigma \leq \rho (x,y)\leq \sigma \}\) is disjoint from C. Thus most compact sets have porosity 1 at any of their points. Similar results on ``thinness'' in various senses hold for continua, curves and graphs of real continuous functions on [0,1]. [See also \textit{A. J. Ostaszewski}, Mathematika 21, 116-127 (1974; Zbl 0305.54040), \textit{J. A. Wieacker}, Math. Ann. 282, 637-644 (1988; Zbl 0636.52004), and the survey of \textit{T. Zamfirescu}, Rend. Semin. Mat., Torino 43, 67-88 (1985; Zbl 0601.52004).]