Open Sets

One of the themes of this (or any other) course in real analysis is the curious interplay between various notions of “big” sets and “small” sets. We have seen at least one such measure of size already: Uncountable sets are big, whereas countable sets are small. In this chapter we will make precise what was only hinted at in Chapter Three – the rather vague notion of a “thick” set in a metric space. For our purposes, a “thick” set will be one that contains an entire neighborhood of each of its points. But perhaps we can come up with a better name.… Throughout this chapter, unless otherwise specified, we live in a generic metric space (M, d).

A set U in a metric space (M, d) is called an open set if U contains a neighborhood of each of its points. In other words, U is an open set if, given x ∈ U, there is some ε > 0 such that Bε(x) ⊂ U.

Examples 4.1

1. (a) In any metric space, the whole space M is an open set. The empty set ø is also open (by default).

2. (b) In ℝ, any open interval is an open set. Indeed, given x ∈ (a, b), let ε = min {x − a, b − x}. Then, ε > 0 and (x − ε, x + ε) ⊂ (a, b).

3. […]