First-countable space

Abstract: Publisher Summary 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.

Abstract: In this paper we introduce a new class of spaces, called W-spaces, which is defined in terms of a simple two-person infinite game. Every first-countable space is a W-space, and every W-space is countably bi-sequential. The W-space property is preserved by subspaces, Σ-products, and open mappings. Separable W-spaces are first-countable. Various other properties of W-spaces are studied, and some questions are posed.

Abstract: . A simple condition on the local bases of a first countable space is shownto imply metrisability, and some new and some well-known metrisation theorems arededuced. Weakening the condition gives new classes of spaces distinct from the classof metrisable spaces. 1. Introduction. It is the primary purpose of this paper to present a simplemetrisation theorem, give its elementary proof and deduce a number of othermetrisation theorems from it. The theorem isTheorem 1. In order that a Tx-space X be metrisable it is necessary and sufficientthat for each x in X, there be a countable decreasing local neighbourhood basis1 {W(n, x): n — 1,2,3,...} atxsatisfying (A) ifxEU and U is an open set, then there exist a positive integer s = s(x, U) and an open set V = V(x, U) containing x such that x E W(s, y) C U whenever y E V. We conclude by considering conditions weaker than (A) and give examples toshow that 'decreasing' cannot be omitted in the statement of Theorem 1. This gives aclass of first countable spaces, distinct from the metrisable spaces, in which everyseparable space is second countable.Throughout, x, y, z will denote elements of a space X, while i,j, m, n, r, s will beelements of the set N of positive integers. For A EX and a family G of subsets of X,St(A,G) will denote the union of those elements of G which meet A, and St(x, G)will denote St({x}, G). X is developable if it has a development, that is, a sequence{G(n): n E N} of open coverings of X such that, for each x, (St(x, G(n)): n E N} isa local basis at x. For A Q X,A° will denote the interior of A,All spaces will satisfy the Tx separation axiom unless we specifically allowotherwise.2. Proof of Theorem 1. The proof of the following lemma is straightforward and isomitted. We shall not require (C) again until the next section, but place it here forconvenience.