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Hyperspaces of a CANR*
Published online by Cambridge University Press: 24 October 2008
Extract
If X is a compact Hausdorff space we denote by S(X), and by C(X), the hyperspaces of X consisting of all non-empty closed sets, and all non-empty connected closed sets. The topology in each case is the finite topology of Michael ((6), Definition 1·7), in which a sub-base for the open sets is taken consisting of all sets of either of the forms {F|F ⊂ G} and {F|F ∩ G ≠ φ} (where G is any open set of X). Michael has shown that S(X) is also compact Hausdorff ((6), Theorem 4·9·6), and S(X) contains in an obvious way sets which are homeomorphic with C(X) and X itself. We recall that if Xis also a metric space, the topology induced on S(X) (and on C(X)) by Hausdorff's metric is the same as the finite topology ((6), Proposition 3·6).
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 4 , October 1961 , pp. 754 - 758
- Copyright
- Copyright © Cambridge Philosophical Society 1961
References
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