CHAPTER 8 - Martin's Axiom and First-Countable S- and L-Spaces

Abstract

This chapter discusses first-countable spaces. Every point in this space has a countable basis for open neighborhoods. If an uncountable regular Hausdorff space does not possess an uncountable discrete subspace, despite the fact that all initial segments in some ω-enumeration of that space are open, then that space is called a right-separated S-space of type ω1. If an uncountable and regular Hausdorff space does not have an uncountable discrete subspace, despite the fact that all final segments in some ω1-enumeration of the space are open, then that space is called a left-separated L-space of type ω1. The chapter further presents some methods for constructing models of Martin's Axiom with some additional properties.