Orthocompactness versus normality in hyperspaces

Abstract

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: • K ( γ ) is orthocompact iff either cf γ ⩽ ω or γ is a regular uncountable cardinal, as a corollary normality and orthocompactness of K ( γ ) are equivalent for every non-zero ordinal γ. 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 K ( S ) of the Sorgenfrey line S is orthocompact therefore so is the Sorgenfrey plane S 2 .