Intervals

We need to be able to talk easily about certain subsets of $\mathbb {R}$. We say that I $\subset$ $\mathbb {R}$ is an open interval if

I = (a, b) = {x $$\in$$ $$\mathbb {R}$$ : a < x < b}.

Thus an open interval excludes its end points, but contains all the points in between. In contrast a closed interval contains both its end points, and is of the form

I = [a, b] = {x $$\in$$ $$\mathbb {R}$$ : a$$\le$$x$$\le$$b}.

It is also sometimes useful to have half - open intervals like (a, b] and [a, b). It is trivial that [a, b] = (a, b) $\cup$ {a} $\cup$ {b}.

The two end points a and b are points in $\mathbb {R}$. It is sometimes convenient to allow also the possibility a = - $\infty$ and b = + $\infty$; it should be clear from the context whether this is being allowed. If these extensions are being excluded, the interval is sometimes called a finite interval, just for emphasis.

Of course we can easily get to more general subsets of $\mathbb {R}$. So (1, 2) $\cup$ [2, 3] = (1, 3] shows that the union of two intervals may be an interval, while the example (1, 2) $\cup$ (3, 4) shows that the union of two intervals need not be an interval.

Exercise 1.7   Write down a pair of intervals I₁ and I₂ such that 1 $\in$ I₁, 2 $\in$ I₂ and I₁ $\cap$ I₂ = $\emptyset$.

Can you still do this, if you require in addition that I₁ is centred on 1, I₂ is centred on 2 and that I₁ and I₂ have the same (positive) length? What happens if you replace 1 and 2 by any two numbers l and m with l$\ne$m?

Exercise 1.8   Write down an interval I with 2 $\in$ I such that 1 $\not\in$I and 3 $\not\in$I. Can you find the largest such interval? Is there a largest such interval if you also require that I is closed?

Given l and m with l$\ne$m, show there is always an interval I with l $\in$ I and m $\not\in$I.