Hyperspaces of countable compacta
. Hyperspaces H ( X ) of all countable compact subsets of a metric space X and A n ( X ) of inﬁnite compact subsets which have at most n ( n ∈ N ), or ﬁnitely many ( n = ω ) or countably many ( n = ω + 1) accumulation points are studied. By descriptive set-theoretical methods, we fully characterize them for 0-dimensional, dense-in-itself, Polish spaces and partially for σ -compact spaces X . Using the theory of absorbing sets, we get characterizations of H ( X ), A ω ( X ) and A ω +1 ( X ) for nondegenerate connected, locally connected Polish spaces X which are either locally compact or nowhere locally compact. For every n ∈ N , we show that if X is an interval or a simple closed curve, A n ( X ) is homeomorphic to the linear space c 0 = { ( x i ) ∈ R ω : lim x i = 0 } with the product topology; if X is a Peano continuum and a point p ∈ X is of order ≥ 2, then the hyperspace A 1 ( X, { p } ) of all compacta with exactly one accumulation point p also is homeomorphic to c 0 .