Hereditary countable metacompactness in finite and infinite product spaces of ordinals

Metacompact subspaces of products of ordinals

“…The first is the investigation of countably metacompact subspaces of the product of ordinals by Kemoto and Smith in [7] and [8]. A synthesis of the main theorems from these papers is proved that a subspace of a linearly ordered space is a D-space iff it is metacompact iff it has no closed subset homeomorphic to a stationary subset of a regular, uncountable cardinal.…”

“…The first is the investigation of countably metacompact subspaces of the product of ordinals by Kemoto and Smith in [7] and [8]. A synthesis of the main theorems from these papers is The second line is an investigation of D-spaces.…”

“…Notions equivalent to κ n -stationary were called stafull in [4] and [5]. Kemoto remarks that even for A ⊂ ω 2 1 , this notion differs from inductively stationary of [8]. Sometimes we will say stationary in place of (κ 1 , .…”

“…. , κ n -stationary = κ n -stationary, clause 1 is in [4] and [5], clauses 1 and the particular case of 2 are in [8], and clause 2 may be new. Note that when n = 1, clause 3 gives the familiar result that an open stationary set contains a final segment.…”

“…To apply κ n -stationary techniques to arbitrary subspaces of κ n (instead of subspaces of κ n ), we need another technique from [8]. We decompose κ n into finitely many pieces, each homeomorphic to κ m for some m ≤ n. For n = 1, κ 1 is κ 1 .…”