Baire category theorem

In the space 〈ℝ, O〉 some sets may be regarded as “thin” and some as “thick.” For example, finite sets and the set ℤ are thin but intervals and open sets are not. The Baire category theory provides a precise definition of these concepts and includes the result that, in a complete metric space, each nonempty open set is a “thick” set.

The historical roots of Baire category theory lie in the last two decades of the nineteenth century and are associated with the characterization of the set Df of discontinuity points of a Riemann integrable function f. The question was, how thick could Df be. It is not perhaps an exaggeration to assert that modern mathematical analysis evolved from two strands of investigations into that question. In one strand, the notion of a measure-zero set (see Definition 7.4.8) was discovered; the other strand started from the notion of a nowhere-dense set (defined below) and sought to find a topological characterization of Df. The first strand led to the creation of modern integration theories and the second gave rise to the subject of point-set topology, which provides a framework for mathematical theories that treat functions as points in an abstract space with geometric and algebraic properties.

The discrete and the continuous are among the most fundamental categories of the human mind, and our urge to create theories that connect the two has prompted us to invent and deploy the infinite set in a monumental intellectual endeavor known as mathematical analysis.

As a branch of mathematics, analysis has evolved during the last four centuries. Prior to this time, mathematics was mainly geometry and arithmetic (together with some algebra). Natural numbers were the primary concepts of arithmetic, which provided for a quantitative study of discrete phenomena; and straight lines, curves, surfaces, etc. were the primary concepts of geometry, which provided for a quantitative study of continuous phenomena. So from a historical point of view, an understanding of the continuous in terms of the discrete could mean none other than constructing analytic models of the primary concepts of geometry using the stuff of arithmetic, which we accomplished under the auspices of our infinite sets.

The familiar real number system is one example of an analytic model of the geometric line, and the familiar system of the complex numbers is one example of an analytic model of the plane.

Introduction Traditional treatments of analysis are based on the ordinary real number system (R, +, ·, <), and their set-theoretical framework is ZFC. In this book, the set-theoretical framework is upgraded to IST and the real number system is upgraded to a system denoted by (ℝ, +, ·, <), where ℝ, +, ·, and < are standard sets in IST. It is useful to think of the system (ℝ, +, ·, <) as one to which the following informal remarks apply.

1. The properties of the elements of (ℝ, +, ·, <) are like the properties of the elements of anultrapower (*R, +, ·, <) of the ordinary real numbers, which we introduced in Definition 0.5.6.

2. The set ℝ, like *R, has both standard and nonstandard elements, and the standard elements of ℝ are none other than the ordinary real numbers. We shall use the symbols σℝ and R interchangeably to denote the class of the standard elements of ℝ. The presence of nonstandard numbers in ℝ endows the system (ℝ, +, ·, <) with external properties.

3. As far as the internal properties of the elements of (ℝ, +, ·, <) are concerned, the theory of (ℝ, +, ·, <) as an entity in IST is formally identical with the theory of (R, +, ·, <) as an entity in ZFC.