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A connected topological space is unicoherent provided that if where and are closed connected subsets of , then is connected. Let be a unicoherent space, we say that makes a hole in if is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.
For a Tychonoff space , let be the family of hypographs of all continuous maps from to endowed with the Fell topology. It is proved that has a dense separable metrizable locally compact open subset if is metrizable. Moreover, for a first-countable space , is metrizable if and only if itself is a locally compact separable metrizable space. There exists a Tychonoff space such that is metrizable but is not first-countable.
We define a metric , called the shape metric, on the hyperspace of all non-empty compact subsets of a metric space X. Using it we prove that a compactum X in the Hilbert cube is movable if and only if X is the limit of a sequence of polyhedra in the shape metric. This fact is applied to show that the hyperspace , dS)2ℝ2
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