Chapter 12 N-Compactness and Its Applications

doi:10.1016/S0924-6509(08)70158-2

North-Holland Mathematical Library

Volume 41, 1989, Pages 459-521

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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.

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Chapter 12 N-Compactness and Its Applications

Publisher Summary

Indag. Math.

General Topology Appl.

Indag. Math.

General Topology Appl.

Indag. Math.

General Topology Appl.

General Topology Appl.

Indag. Math.

Topological Algebras

Boolean rings of projection maps

J. London Math. Soc.

Spaces in which special sets are z-embedded

Canad. J. Math.

On μ-spaces and kR-spaces

Proc. Amer. Math. Soc.

N-compactness and weak homogeneity

Proc. Amer. Math. Soc.

Another realcompact, 0-dimensional, non-N-compact space

Proc. Amer. Math. Soc.

Homological Algebra

On the Hewitt realcompactification of a product space

Trans. Amer. Math. Soc.

Normality in subsets of product spaces

Amer. J. Math.

A Boolean power and a direct product of abelian groups

Tsukuba J. Math.

On abelian groups of integer-valued continuous functons, their Z-duals and Z-reflexivity

General Topology

Spaces of continuous functions (V)

Studia Math.

Rings of Continuous Functions

Banach spaces

Bull. Soc. Math. France, Mém.

E-kompakte Räume

Math. Z.

Categorical topology 1971–1981

Rings of real-valued continuous functions, I

Trans. Amer. Math. Soc.

The additive group of commutative rings generated by idempotents

Proc. Amer. Math. Soc.

Introduction to Set Theory

The Hahn-Banach theorem for non-Archimedean valued fields

Proc. Cambridge Phil. Soc.

Set Theory

Measures in fully normal spaces

Fund. Math.

Equivalence of a problem in measure theory to a problem in the theory of vector lattices.

Bull. Amer. Math. Soc.

On μ-spaces and k_R-spaces