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Comparison of Some Classical Topologies on Hyperspaces within the Framework of Point Spaces
Authors:Meriem Arab and Jean Calbrix
Abstract:S. Dolecki, G. Greco and A. Lechicki call a space X consonant if the co-compact topology and the upper Kuratowski topology on the set of closed subsets of X coincide. We call a space X hyperconsonant if Fell's topology and the (Kuratowski) convergence topology coincide. Recently, we proved that a first countable, locally paracompact, T3-space is hyperconsonant if and only if the space possesses at most one point without a compact neighbourhood, extending the same result of D. Fremlin obtained for metrizable spaces. In this paper, we pursue the study of hyperconsonance within the framework of point spaces (countable T1-spaces with exactly one accumulation point) and we compare consonance and hyperconsonance in such spaces. In particular, we answer a question of T. Nogura and D. Shakhmatov: does there exist a nonconsonant point space? We provide a Fréchet, 
$$aleph _0 $$
-point space which is not consonant. Moreover, this example proves that the consonance is not preserved by continuous closed compact-covering maps of separable complete metrizable spaces onto Hausdorff spaces.
Keywords:hyperspaces  Fell's topology  (Kuratowski) convergence topology  consonance  hyperconsonance  point spaces
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