Well ordered sequences

Abstract

The following theorem is equivalent to the Axiom of Choice. Theorem. Each partially ordered set (P,<) has a well ordered cofinal subset (S, precedes) such that precedes is compatible with < on S; i.e., if a,b in S with a< b then a precedes b. (Note: a succeeds b does not imply a< b.) Corollary. Each point in a topological space has a well ordered local base such that the well ordering is compatible with the partial ordering of set inclusion. A well ordered sequence (WOS) is a function psi:E to X from a well ordered set without a last point into a space X. A well ordered subsequence (WOSS) of psi, is psi restricted to a cofinal subset of E. Psi (converges, quasi-converges) to p in X if for each neighborhood U of p there is a (residual, cofinal) subset S of E with psi(S) subset U. Quasi-convergence classes (QC classes) are also defined. Theorem. Each QC class in a set X determines a topology on X and conversely, each topology on X determines a QC class. A WOS psi:E to X is said to (biconverge, quasi-biconverge) to two points p and q if for each pair of neighborhoods U and V of p and q respectively there exists a (residual, cofinal) subset S of E such that psi(S) subset U intersection V. Theorem. A space is Hausdorff if and only if whenever a WOS quasi-biconverges to two points p and q then p = q. Theorem. A space is compact if and only if each WOS quasi-converges. Theorem. Lindelof and the property that each WOS with no countable WOSS quasi-converges are equivalent in countably metacompact spaces. Theorem. A mapping f:X to Y is pseudo-open (Arhangel'skii) if and only if for each WOS psi subset Y that quasi-converges to some p in Y there is a WOS xi subset f^-1(psi) that quasi-converges to some q in f (p). Similar characterizations of open, closed, continuous, and quotient mappings are also given. A space X is said to be C_1 if no WOS in X-U converges to a point of U subset X implies U is open. X is called C_2 if whenever p in Cl(F) there is a WOS in F that converges to p. M pseudo-open mappings are defined in terms of quasi-convergence. Theorem. A space Y is C_1 if and only if it is the quotient of a CLB space and is C_2 if and only if it is the continuous M pseudo-open image of a CLB space where M is the cardinality of the power set of Y. (CLB stands for chain local base.) A space is said to be accumulation complete if whenever a sequence accumulates (quasi-converges) to some point there exists a subsequence that converges to this point. Accumulation complete spaces are of interest not only because they generalize first countable spaces but because they provide an answer to the question of Franklin asking ''When is a sequential space Frechet?" Countably compact accumulation complete spaces are also sequentially compact. M accumulation complete spaces and strong M pseudo-open mappings are also defined for each cardinal M. Theorem. A space is accumulation complete if and only if it is the Aleph-zero pseudo-open image of a first countable space. Theorem. A space is M accumulation complete if and only if it is the strong M pseudo-open image of an M CLB space.