Hyperspaces of finite subsets as boundary sets (Q1064561)

Let X denote a connected, locally path-connected, \(\sigma\)-compact metric space. \({\mathcal F}(X)\) is the hyperspace of all nonempty finite subsets of X, topologized by the Hausdorff metric. Let \({\mathcal E}\) denote a \(\sigma\)- compact subspace of \({\mathcal F}(X)\) with the property that, for \(E\in {\mathcal E}\) and \(F\in {\mathcal F}(X)\) with \(E\subset F\), \(F\in {\mathcal E}\). If X admits a Peano compactification \(\bar X,\) then \({\mathcal E}\) is a \(\sigma\) Z-set in its closure \(\bar {\mathcal E}\) in the hyperspace \(2^{\bar X}\), and \(\bar {\mathcal E}\) is a topological Hilbert cube. We show that \({\mathcal E}\) contains an fd-cap set (and is therefore a boundary set) for \(\bar {\mathcal E}\) if and only if the remainder \(\bar X\setminus X\) is locally non-separating in \(\bar X.\) In particular, if \(X=\bar X\) is a Peano continuum, then \({\mathcal F}(X)\) is a boundary set for \(2^ X\).