Orthocompactness versus normality in hyperspaces

摘要

For a regular space X, 2~X denotes the collection of all non-empty closed sets of X with the Vietoris topology and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of 2~x. In this paper, we will prove: Κ(γ) is orthocompact iff either cf γ≤ω or γ is a regular uncountable cardinal, as a corollary normality and orthocompactness of Κ(γ) are equivalent for every non-zero ordinal y. We present its two proofs, one proof uses the elementary submodel techniques and another does not. This also answers Question C of Kemoto (2007) [4], Moreover we discuss the natural question whether 2~ω is orthocompact or not. We prove that 2~ω is orthocompact iff it is countably metacompact, the hyperspace Κ(S) of the Sorgenfrey line S is orthocompact therefore so is the Sorgenfrey plane S~2.