Normality and compactness are equivalent in hyperspaces
James Edgar Keesling · Bulletin of the American Mathematical Society · 1970
Introduction.Let Ibea Hausdorff topological space and 2 X the space of all closed subsets of X with the finite topology [5, Definition 1.7, p. 153] or [4].This topology is also known as the exponential or Vietoris topology and 2 X is known as the hyperspace of X.It is known that 2 X is regular if and only if 2 X is completely regular if and only if X is normal [5, Theorem 4.9, p. 163].When is 2 X normal?It is clear that 2 X is normal if X is compact for then 2 X is compact Hausdorff.Is the converse true?Ivanova has shown in [2] that if X is a well ordered space with the order topology, then 2 X normal implies that X is compact.In [3] the author has shown that 2 2 is normal if and only if X is compact.The purpose of this announcement is to indicate that, assuming CH (the continuum hypothesis), 2 X normal implies that X is compact.