Lattices, Filters, and Topological Spaces

Abstract

There is a close relation between partially ordered sets and topological spaces. The theme of this chapter is a development of this relationship. In a first course in topology, orders are often used to develop important tools such as filters and nets and to define a topology on some spaces, such as the space of real numbers. In this chapter, we start by defining orders and discussing several of the ways in which they are associated with spaces. In the middle of the chapter, we introduce spaces whose topologies are defined by an order, and prove that such spaces are hereditarily normal. The chapter closes with a summary of the properties of well-ordered sets and ordinal numbers, and an investigation of the order topology on an ordinal number.