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Characterization of knot complements in the n-sphere

Vo-Thanh Liem, Gerard Venema (1995)

Fundamenta Mathematicae

Knot complements in the n-sphere are characterized. A connected open subset W of S n is homeomorphic with the complement of a locally flat (n-2)-sphere in S n , n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of S 1 in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema.

Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

T. Banakh, R. Voytsitskyy (2008)

Colloquium Mathematicae

It is shown that the hyperspace C l d H ( X ) (resp. B d d H ( X ) ) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

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