Typical elements and meager spaces (Q1398189)

Let \(M\) be a complete metric space. A subset \(A\) of \(M\) is called nowhere dense if its closure \(A\) contains no open subset. A subset is called meager if it is the countable union of nowhere dense subsets. The complement of a meager set is called a generic set. An element of a generic set is called typical. Let \(\mathbb{K}^n\) be the set of all compact subsets of \(\mathbb{R}^n\) equipped with the Hausdorff metric. For fixed \(\varepsilon> 0\), let \(D^n_\varepsilon\) be the set of all \(X\in\mathbb{K}^n\) such that each component of \(X\) contains an open ball of radius \(\varepsilon\). The author characterizes the typical element of \(D^n_\varepsilon\), as below: The ball-and-chain theorem. For \(n\geq 2\), the typical element of \(D^n_\varepsilon\) is the union of a closed \(\varepsilon\)-ball and a pseudo-arc, and as a consequence, the Hausdorff dimension of its boundary equals \(n-1\). The author also proves several subspaces of \(\mathbb{K}^n\) to be meager. Theorem 2. For \(n\geq 2\), the following subspaces of \(\mathbb{K}^n\) are each the union of a countable number of nowhere dense sets. (1) The space \(D'\), in which each element consists of a finite number of connected components; the \(i\)th component contains an open ball of radius \(\varepsilon\). (2) The subspace of all compact \(n\)-dimensional submanifolds of \(\mathbb{R}^n\). (3) The subspace \(M\) of all \(n\)-dimensional compact submanifolds of \(\mathbb{R}^n\) having diameter greater than some preassigned \(\varepsilon\). (4) The subspace \(M'\) of all \(n\)-dimensional compact connected submanifolds of \(\mathbb{R}^n\) that have diameter greater than some preassigned \(\varepsilon\). Theorem 3. The following two subspaces of \(\mathbb{K}^n\) are meager. (1) The subspace \(P\), which consists of all compact, connected and locally connected sets. (2) The subspace AC, which consists of all arc-connected compact subsets.