Chapter 12 N-Compactness and Its Applications

Abstract

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.