Isaac Namioka

σ-fragmentable Banach spaces

§1. Introduction . Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X . The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ -diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each e>0, it is possible to write where each set X i , i ≥1, has the property that each non-empty subset of X i , has a non-empty relatively open subset of ρ -diameter less than e. If is any family of subsets of X , we say that X is σ-fragmented by the metric ρ , using sets from , if, for each e>0, the sets X i , i ≥ 1, in (1.1) can be taken from

Sigma-fragmentability of mappings into Cp(K)

Abstract We prove the main theorem concerning the σ-fragmentability of a continuous map ϑ : T → C p ( K ), where T is a topological space and K is a compact Hausdorff space, and we obtain from it some new and old results about the σ-fragmentability of subsets of C p ( K ) and property N ∗f . Here are a few of them: A subset T of C p ( K ) is σ-fragmented by the norm metric if and only if it is cover-semicomplete, i.e., T is fragmented by a pseudo-metric d such that each d -convergent sequence in T has a cluster point in T . The class of cover-semicomplete spaces contains spaces with countable separation in the sense of Kenderov and Moors. Assume that a compact space K has a family A of compact subsets such that (a) for each A e A , C p ( A ) is σ-fragmented by the norm, (b) if A n e A for each n e N , then ∪ n = 1 ∞ A n is compact and (c) ∪A is dense in K . Then C p ( K ) is σ-fragmented by the norm. Finally, suppose K and L are compact spaces and L has property N ∗ . If C p ( K ) embeds in C p ( L ), then K has property N ∗ . The last result can be used to show that C p ( βN ) does not embed in C p (0, 1 Γ ) for any set Γ. The method of proving the main theorem also yields the following. If K and L are compact and C p ( K ) and C p ( L ) are σ-fragmented by the norm, so is C p ( K × L ).

Furstenberg's structure theorem via CHART groups

We give an almost self-contained group theoretic proof of Furstenberg’s structure theorem as generalized by Ellis: Each minimal compact distal flow is the resul t of a transfinite sequence of equicontinuous extensions, and their limits, starting from a flow consistin g of a singleton. The groups that we use are CHART groups, and their basic properties are recalled at the beginning of this paper.

Continuous functions on products of compact Hausdorff spaces

In [4], we investigated the spaces of continuous functions on countable products of compact Hausdorff spaces. Our main object here is to extend the discussion to arbitrary products of compact Hausdorff spaces. We prove the following theorems in Section 3.

Fragmentability in banach spaces: Interaction of topologies

Let (X, τ) be a topological space and let ρ be a metric on X. Then one occasionally encounters the following situation: For each ε > 0 and a non-empty subset A ⊂ X, there exists a τ-open subset U of X such that U ∩ A = ϕ and ρ-diameter of U ∩ A is less than ∈. If this is the case then (X, \gt) is said to be fragmented by \gr. For instance a weakly compact subset of a Banach space with the weak topology is fragmented by the norm metric, and this fact has many consequences. For non-compact spaces, the natural analog of fragmentability is \gs-fragmentability. In this exposition, these two notions are examined and their applications are described.ResumenCuando en un espacio topológico (X, τ) tenemos además una métrica ρ, a veces la siguiente situación se presenta: para cada ∈ > 0 y para cada conjunto A ⊂ X, existe un subconjunto τ-abierto U de X tal que U ∩ A = ϕ y ρ-diámetro de U ∩ A es menor que ∈. Si este último es el caso, (X, τ) se dice que está fragmentado por ρ. Por ejemplo, los subconjuntos débilmente compactos de un espacio de Banach con su topología débil están fragmentados por la métrica asociada a la norma: este resultado tiene muchas consecuencias. Para espacios que no son compactos, el análogo natural de la noción de fragmentabilidad es la noción σ-fragmentabilidad. En este artículo expositivo, analizamos las nociones de fragmentabilidad y σ-fragmentabilidad así como aplicaciones de las mismas.

The Category Theorems

This short chapter is concerned with the concept of category and with its application to the theory of linear topological spaces. The results include some of the most profound and most useful theorems of the subject of linear topological spaces, and are among the most important of the applications of category.

Convexity in Linear Topological Spaces

This chapter, which begins our intensive use of scalar multiplication in the theory of linear topological spaces, marks the definite separation of this theory from that of topological groups. The results obtained here do not have generalizations or even analogues in the theory of groups.