Haruto Ohta

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.

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, settheoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of ωi consisting of ordinals of cofinality ω for each 1 < i < ω, then there exist abelian groups An(n ∈ Z) such that An ∼= An+1 and An 6∼= An+2 for each n ∈ Z. (2) There exist abelian groups An(n ∈ Z) such that An ∼= An+1 for each n ∈ Z and An 6∼= An+2 for each n < 0. The groups An are the groups of Z-valued continuous functions on a topological space and their dual groups.

Chapter 12 N-Compactness and Its Applications

Publisher Summary This chapter discusses N -compactness and its application and N -compact spaces. A real-compact space is a topological space that can be embedded in the product of copies of the real-line ℝ as a closed subspace. A topological space is called “ N -compact,” if it is homeomorphic to a closed subspace of the product of copies of the countable discrete space N . The chapter also introduces the notion of E -compact spaces, rings and lattices of continuous functions, and applications to Abelian groups and non-Archimedean Banach spaces. N -compact spaces can be regarded as a 0-dimensional analog of real-compact spaces. The chapter considers the relationship between real-compact spaces and N -compact spaces and gives technical results and examples. The chapter explains that the ring structure or the lattice structure of C ( X , ℤ) determines the topology of an N -compact space X , where C ( X , ℤ) is the ring or the lattice of integer-valued, continuous functions on X . Applications to Abelian groups, where N -compactness will play an important role to reduce the reflexivity of Abelian groups. Banach spaces over certain non-Archimedean-valued fields have many features on the reflexivity that are similar to those of Abelian groups. Applications to non-Archimedean Banach spaces apply N -compact spaces to such Banach spaces, with particular attention paid to their similarity.

Prime subspaces in free topological groups

Abstract Let F(X) (A(X)) be the free (Abelian) topological group over X. We prove: If P is one of the spaces R , Q , R , Q , βω, βω ω and 2/gk for an infinite κ and if F(X) or A(X) contains a copy of P, then X contains a copy of P. If P is the one-point compactification of an infinite discrete space or ω1 + 1, this is not true. If P = ω1, this holds for F(X) but is independent of ZFCfor A(X).

Abelian groups of continuous functions and their duals

Abstract Let A ∗ = Hom (A, Z ) for an Abelian group A, were Z is the group of integers. A∗ is endowed with the topology as a subspace of Z A. Then, for a 0-dimensional space X and an infinite cardinal κ the following are equivalent. (1) There exists a free summand of C(X, Z ) of rank κ; (2) there exists a subgroup of C(X, Z )∗ isomorphic to Z κ; (3) there exists a compact subset K of β N X with w(K)⩾κ; (4) there exists a compact subset K of C(X, Z )∗ with w(K)⩾κ. There exist groups A such that A∗ is a subgroup of Z N and A∗ is not isomorphic to A∗∗∗.

Extensions of zero-sets in the product of topological spaces

Abstract This paper studies when, for completely regular, Hausdorff spaces X and Y , ( z ) the product X × Y is z -embedded in X × βY . For various classes P of spaces, those spaces Y such that ( z ) for each X e P and those spaces X such that ( z ) for each Y e P are determined. It is also proved that X is a metric space iff X × Ye 0 z for each Y e 0z.

Topological spaces whose baire measure admits a regular borel extension

A completely regular, Hausdorff space X is called a Malik space if every Baire measure on X admits an extension to a regular Borel measure. We answer the questions about Malik spaces asked by Wheeler [29] and study their topological properties. In particular, we give examples of the following spaces: A locally compact, measure compact space which is not weakly Bairedominated; i.e., it has a sequence Fn I 0 of regular closed sets such that nnfl, Bn 74 0 whenever Bn s are Baire sets with Fn C Bn; a countably paracompact, non-Malik space; a locally compact, non-Malik space X such that the absolute E(X) is a Malik space; and a locally compact, Maiik space X for which E(X) is not. It is also proved that Michaels product space is not weakly Baire-dominated.

Extension problems of real-valued continuous functions

Publisher Summary This chapter discusses extension problems of real-valued continuous functions. In this discussion, a space means a completely regular T 1 -space. A subset A of a space X is said to be C -embedded in X if every real-valued continuous function on A extends continuously over X , and A is said to be C * -embedded in X if every bounded real-valued continuous function on A extends continuously over X . Every C*-embedded subset of a first countable space is closed. Kulesza–Levy–Nyikos has proved that if b = s = c , then there exists a maximal almost disjoint family ℛof infinite subsets of ℕ such that every countable set of nonisolated points of the space ℕ Uℛ is C*-embedded. Because every set of nonisolated points of ℕ Uℛ is discrete and ℕ Uℛ is pseudocompact, those countable sets are not C -embedded. This chapter aims to collect some open questions concerning C -, C * -embeddings and extension properties which can be described by extensions of real‑valued continuous functions. Details of π -embedding and π Ƶ -embeddings are also provided.

Local compactness and Hewitt realcompactifications of products II

Abstract We generalize and refine results from the authors paper [18]. For a completely regular Hausdorff space X, υX denotes the Hewitt realcompactification of X. It is proved that if υ(X×Y)=υX×υY for any metacompact subparacompact (or m-paracompact) space Y, then X is locally compact. A P(n)-space is a space in which every intersection of less than n open sets is open. A characterization of those spaces X such that υ (X×Y = υX×υY for any (metacompact) P(n)-space Y is also obtained.

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

A metric space X has the unique midset property if there is a topology-preserving metric d on X such that for every pair of distinct points x , y there is one and only one point p such that d ( x , p ) = d ( y , p ). The following are proved. (1) The discrete space with cardinality [nfr ] has the unique midset property if and only if [nfr ] ≠ 2, 4 and [nfr ] [les ] [cfr ], where [cfr ] is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than [cfr ], then the countable power D N of D has the unique midset property. In particular, the Cantor set and the space of irrational numbers have the unique midset property. A finite discrete space with n points has the unique midset property if and only if there is an edge colouring ϕ of the complete graph K n such that for every pair of distinct vertices x , y there is one and only one vertex p such that ϕ( xp ) = ϕ( yp ). Let ump( K n ) be the smallest number of colours necessary for such a colouring of K n . The following are proved. (3) For each k [ges ] 0, ump( K 2 k +1 ) = k . (4) For each k [ges ] 3, k [les ] ump( K 2 k ) [les ] 2 k −1.