Chapter 16 - Perfect compacta and basis problems in topology

Abstract

This chapter discusses the concept of perfect compacta and some basis problems in topology. An interesting example of a compact Hausdorff space that is often presented is the unit square [0, 1]×[0, 1] with the lexicographic order topology. The closed subspace consisting of the top and bottom edges is perfectly normal. This subspace is often called the Alexandroff double arrow space. It is also sometimes called the “split interval”, because it can be obtained by splitting each point x of the unit interval into two points x0, x1, and defining an order by declaring x0 <x1 and using the induced order of the interval otherwise. The top edge of the double arrow space minus the last point is homeomorphic to the Sorgenfrey line, as is the bottom edge minus the first point. Hence it has no countable base, so being compact, is non-metrizable. There is an obvious two-to-one continuous map onto the interval. The concepts of uncountable spaces are also discussed in the chapter. A discussion on different approaches and axiomatics is also presented.