In this chapter, we consider some particular sorts of subsets of R, and their relation to convergence. This involves many definitions; familiarity will only come with use. We study the ideas that arise here in a more general setting in Volume II.

Closed sets

We begin by considering intervals in R. A subset I of R is an interval if whenever two numbers belong to it, then so do all the intermediate points: that is, if a < c < b and a, b ∈ I then c ∈ I. R is an interval. The empty set and singleton sets are degenerate intervals. Other examples of intervals are the semi-infinite intervals

(−∞,b)= {x ∈ R : x < b}, (−∞,b]= {x ∈ R : x ≤ b},

(a, ∞)= {x ∈ R : a < x}, [a, ∞)= {x ∈ R : a ≤ x},

and the bounded intervals

(a, b)=(b, a)= {x ∈ R : a < x < b},(a, b]=[b, a)= {x ∈ R : a < x ≤ b},

[a, b)=(b, a]= {x ∈ R : a ≤ x < b},[a, b]=[b, a]= {x ∈ R : a ≤ x ≤ b},

where a < b. It is an easy exercise to show that every interval is of one of these forms. The length of a bounded interval is b − a; the length of R and of semi-infinite intervals is +∞.