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The Integers and Topology

1984; Elsevier BV; Linguagem: Inglês

10.1016/b978-0-444-86580-9.50006-9

Eric K. van Douwen,

semigroups and automata theory

This chapter discusses integers and topology. Role in topology of certain cardinals is associated with ω. This chapter discusses the problems on first countability, convergence, and separable metrizable spaces. A typical use of these set theoretic cardinals associated with ω involves topologically defined cardinals. Another use of these set theoretic cardinals associated with ω is that certain topological results hold if one of these cardinals equals ω1. The chapter also discusses the set theory, which states an ordinal is the set of smaller ordinals, and a cardinal is an initial ordinal. ω is ω0, and c is 2ω. The chapter also describes sequential and countable compactness. A countable set A of a space X is said to cluster at x ∈ X if each neighborhood of x contains infinitely many points of A, and it is said to converge to x ∈ X if each neighborhood of x contains all but finitely many points of A. A space is called countably compact if each countably infinite set clusters at some point, and it is called sequentially compact if each countably infinite set has an infinite subset that converges somewhere. Moreover, a space X is called subsequential if for every countably infinite A ⊆ X and for every cluster point x of A, there is an infinite subset of A that converges to x.