Nobuyuki Kemoto

Products of spaces of ordinal numbers

Abstract Let A and B be subspaces of the initial segment of an uncountable ordinal number κ with the order topology. We prove: 1. (i) For the product A × B, the following properties (1)-(3) are equivalent: (1) shrinking property; (2) collectionwise normality; and (3) normality. 2. (ii)For the product A × B, the following properties (4)-(7) are equivalent: (4) strong D- property; (5) expandability; (6) countable paracompactness; and (7) weak D(ω)-property. 3. (iii) If κ = ω1, then for A × B, the properties (1)-(7) above are equivalent, and A × B has one of them iff A or B is not stationary, or A ∩ B is stationary. 4. (iv) If κ is regular and A and B are stationary, then A × Bis κ-compact iff A ∩ B is stationary.

Linearly ordered extensions of GO spaces

Abstract We prove a main theorem: Theorem. There always exists a minimal linearly ordered d -extension of a GO space, where a LOTS Y is said to be a linearly ordered d -extension of a GO space 〈 X , τ, ≤〉 if Y contains X as a dense subspace and the ordering of Y extends the ordering ≤ of X . As some applications of the Theorem, (1) we give a partial negative answer to a problem: “Does every perfect GO space have a perfect orderable d -extension?” (2) For a discrete space 〈 X , τ〉 of cardinality ω 1 , there is a linear ordering ≤ of X such that 〈 X , τ, ≤〉 is a GO space and whose every linearly ordered d -extension contains an order preserving copy of the ordinal space ω 1 as a dense subspace.

Orthocompactness and normality of products with a cardinal factor

Abstract Let κ be a regular uncountable cardinal. In this paper, we introduce two new cardinal functions. Using the first one, we characterize metacompact spaces whose products with κ are orthocompact. This deduces that the orthocompactness of such products imply their normality. The second one is a modification of tightness, and using it, we also characterize compact spaces whose products with κ are normal.

Hereditary countable metacompactness in finite and infinite product spaces of ordinals

Abstract We show ω 1 n is hereditarily countably metacompact for each n ϵ ω , but ω 1 ω is not.

The product of two ordinals is hereditarily countably metacompact

Abstract It will be shown that μ × ν is hereditarily countably metacompact for any ordinals μ and ν . As an immediate corollary we see that ω 1 2 is hereditarily countably metacompact. This answers a question of Ohta ( K. Tamano, 1995 ). Also, as a corollary we see that if A and B are subspaces of ordinals, then A × B is countably metacompact. This corollary answers Question ( iii ) of N. Kemoto et al. (1992, p. 250) .

Countable paracompactness versus normality in ω21

Abstract We will see that: (1) In ZFC, for each subspace X⫅ω 2 1 , the following are equivalent; (a) X is normal, (b) X is countably paracompact and strongly collectionwise Hausdorff, (c) X is expandable. (2) Under a variety of different set-theoretic assumptions (including V = L and PMEA) all countably paracompact subspaces of ω 2 1 are normal. (3) All subspaces of ω 2 1 are collectionwise Hausdorff.

Strong zero-dimensionality of products of ordinals

Abstract We show that the product of finitely many subspaces of ordinals is strongly zero-dimensional. In contrast, for each natural number n, there is a subspace of (ω+1)× c of dimension n.

Generalized paracompactness of subspaces in products of two ordinals

Abstract We will characterize metacompactness, subparacompactness and paracompactness of subspaces of products of two ordinal numbers. Using them we will show: 1. For such subspaces, weak submetaLindelofness, screenability and metacompactness are equivalent. 2. Metacompact subspaces of ω 1 2 are paracompact. 3. Metacompact subspaces of ω 2 2 are subparacompact. 4. There is a metacompact subspace of ( ω 1 +1) 2 which is not paracompact. 5. There is a metacompact subspace of ( ω 2 +1) 2 which is not subparacompact.

Subnormality in ω12

Abstract A space X is said to be subnormal (= δ - normal ) if every pair of disjoint closed sets can be separated by disjoint G δ -sets. It is known that the product space ( ω 1 +1)× ω 1 is neither normal nor subnormal, moreover the subspace A × B of ω 1 2 is not normal whenever A and B are disjoint stationary sets in ω 1 . We will discuss on subnormality of subspaces of ω 1 2 .

Normality and paranormality in product spaces

Abstract Let βω denote the Stone–Cech compactification of the countable discrete space ω . We show that if p is a point of βω ⧹ ω , then all subspaces of ( ω ∪{ p })× ω 1 are paranormal, where ( ω ∪{ p }) is considered as a subspace of βω . This answers a van Douwens question. Moreover we show that the existence of a paranormal non-normal subspace of ( ω +1)× ω 1 is independent of ZFC, where ω +1 is the ordinal space {0,1,2,…, ω } with the usual order topology.