Vol. 33, No. 3, 1970

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On the equivalence of normality and compactness in hyperspaces

James Edgar Keesling

Vol. 33 (1970), No. 3, 657–667

Let X be a topological space and 2X the space of all closed subsets of X with the finite topology. Assaming the continuum hypothesis it is shown that 2X is normal if and only if X is compact. It is not known if the continuum hypothesis is a necessary assumption, but it is shown that for X a k-space, 2X normal implies X compact. A theorem about the compactification of the n-th symmetric product of a space X is first proved which then plays an important part in the proof of the above results.

Mathematical Subject Classification
Primary: 54.20
Received: 26 September 1969
Revised: 5 December 1969
Published: 1 June 1970
James Edgar Keesling