Orthocompactness versus normality in hyperspaces

For a regular space X, X2 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 X2. In this paper, we will prove:•K(γ) is orthocompact iff either 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 ω2 is orthocompact or not. We prove that•ω2 is orthocompact iff it is countably metacompact,•the hyperspace K(S) of the Sorgenfrey line S is orthocompact therefore so is the Sorgenfrey plane S2.