# zbMATH — the first resource for mathematics

Hyperspaces of finite subsets as boundary sets. (English) Zbl 0575.54009
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 $$2^ X$$.

##### MSC:
 54B20 Hyperspaces in general topology 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54F65 Topological characterizations of particular spaces 57N20 Topology of infinite-dimensional manifolds
Full Text:
##### References:
 [1] R.D. Anderson, On sigma-compact subsets of infinite-dimensional spaces, unpublished manuscript. [2] M. Bestvina, P. Bowers, J. Mogilski, and J. Walsh, Characterization of Hilbert space manifolds revisited, Topology Appl., submitted. [3] Chapman, T.A, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. amer. math. soc., 154, 399-426, (1971) · Zbl 0208.51903 [4] Curtis, D.W, Growth hyperspaces of Peano continua, Trans. amer. math. soc., 238, 271-283, (1978) · Zbl 0344.54009 [5] Curtis, D.W, Hyperspaces of noncompact metric spaces, Comp. math., 40, 139-152, (1980) · Zbl 0431.54004 [6] Curtis, D.W, Boundary sets in the Hilbert cube, Topology and its applications, 20, 3, 201-221, (1985) · Zbl 0575.57008 [7] Curtis, D; Dobrowolski, T; Mogilski, J, Some applications of the topological characterizations of the sigma-compact spaces l2f and σ, Trans. amer. math. soc., 284, 837-846, (1984) · Zbl 0563.54023 [8] Curtis, D; Nhu, N.T, Hyperspaces of finite subsets which are homeomorphic to ℵ_0-dimensional linear metric spaces, Topology and its applications, 19, 3, 251-260, (1985) · Zbl 0587.54015 [9] Mogilski, J, Characterizing the topology of infinite-dimensional σ-compact manifolds, Proc. amer. math. soc., 92, 111-118, (1984) · Zbl 0577.57005 [10] Torunczyk, H, Skeletonized sets in complete metric spaces and homeomorphisms of the Hilbert cube, Bull. acad. polon. sci., 18, 119-126, (1970) · Zbl 0202.54003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.