Kaori Yamazaki

A cardinal generalization of C∗ -embedding and its applications

Abstract As for extending real-valued continuous functions or continuous pseudometrics on a subspace to the whole space, notions of z -, C ∗ -, C -, P - and Pγ -embeddings are known. As a cardinal generalization of z -embedding, Blair defined in 1985 the notion of zγ -embedding with γ≥ω , where zω -embedding coincides with z -embedding. On the other hand, since Pω -embedding equals C -embedding, Pγ -embedding can be also regarded as a cardinal generalization of C -embedding. Recently Ohta asked if a cardinal generalization of C ∗ -embedding can be defined so that this property plus Uω -embedding is equal to Pγ -embedding, and it is itself equals C ∗ -embedding in case γ=ω . In this paper, we give a cardinal generalization of C ∗ -embedding, called (P ∗ ) γ -embedding, and answer this problem. As a characterization of (P ∗ ) γ -embedding, we show that (P ∗ ) γ -embedding naturally admits its description by using continuous maps from a subspace into the hedgehog with γ spines. We also give a new extension-like property called weak zγ -embedding, with which z - (respectively C ∗ -, C - or Uω -) embedding equals zγ - (respectively (P ∗ ) γ -, Pγ - or Uγ -) embedding.

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.

Extensions of functions on product spaces

Abstract Let X be a topological space and A its subspace. The following problem posed by Przymusinski in 1983 remains open: for a nondiscrete metric space Y is it true that A × Y is C ∗ - embedded in X × Y iff A × Y is C-embedded in X × Y? In this paper first we prove that this problem is affirmative in case of nondiscrete metrizable Y with Y = Y2, which contains therefore the cases that Y is the Baire zero-dimensional metric space B(κ), Hilbert space R ℵ 0 and J(κ) ℵ 0 , where J(κ) is a hedgehog of κ spines. Secondly, we discuss some results of equivalence between C ∗ - embedding (or C-embedding) and P-embedding of A × Y in X × Y in case of Y being a paracompact Σ- or σ-space.

Products of Weak P -Spaces and K -Analytic Spaces

Let κ be an infinite cardinal. Okuyama showed that the product space X ×i Y of a paracompact weak P (ω)-space X and a K -analytic space Y is paracompact. In this paper, by using the notion of κ- K -analytic spaces which is basically defined by Hansell, Jayne and Rogers, the above result is extended and some other results are given related to normality, collectionwise normality and covering properties on products. An answer to a question of Okuyama and Watson is also given, as well as some applications to extensions of continuous functions on these products.

P(locally-finite)-embedding and rectangular normality of product spaces

Let γ and κ be infinite cardinals and λ a cardinal. A subspace A of a space X is said to be Pγ(locally-finite)-embedded in X if every locally finite partition of unity with cardinality ⩽γ on A can be extended to a locally finite partition of unity on X; this extension property was defined by Dydak recently. In this paper, introducing a space Jγ(κ) and a class of spaces called spaces of type t(γ,κ,λ), we characterize Pγ(locally-finite)-embedding by products with these spaces. We also characterize extendability of locally finite κ+-open covers U of a closed subspace A of a normal space X to those of X by products with these spaces; this extends Przymusinskis result in 1984 of the case |U|=ω. We apply this result to give characterizations of functionally Katětov spaces and Katětov spaces by rectangular normality of products with these spaces.