What is the length of an open interval and how do you prove it?

Question

If an open interval does not contain its endpoints, why does its length is the same as a closed interval with the same endpoints? For example, $d((3,6)) = d([3,6])$.

Answer 1

The length of an interval $(a,b)$ (or $[a,b]$, or $(a,b]$, or $[a,b)$) is defined to be $b-a$. This is just a definition, so it requires no proof. Intuitively, it should make sense: if you think of the interval as being a "stick" cut out from the number line, then it would be $b-a$ units long. We don't care whether the interval contains its endpoints because a single point has no length.

The following intuition may help. Suppose you have a ruler, with lengths marked in centimeters along it. If you want to measure a length of $c$ centimeters, you would usually just measure from the start of the ruler to the point marked $c$. However, you could also measure from a point marked $a$ to a point marked $b$, as long as $b-a=c$. The reason is that you could just shift the ruler forwards by $a$ centimeters, so the point that was marked $a$ is now at the start of the ruler and the point that was marked $b$ is now marked $b-a$.

A vast generalization of this notion of "length of an interval" is Lebesgue measure on $\mathbb{R}$, which is a way of defining the "length" of much more complicated sets than just an interval. In the context of Lebesgue measure, depending on your definitions, it may be a theorem that the length of $(a,b)$ is $b-a$. But in calculus or basic analysis, this is usually just taken as a definition.

Answer 2

This question is more related to measure theory than topology. The most standard measure on the collection of sets generated by open intervals is Lebesgue measure where $\mu([a,b])=b-a$. You can associate a measure with any right continuous function F such that $\mu_F([a,b])=F(b)-F(a)$. There's no correct answer for the measure of an interval, there's lots of different ways to do it.

Answer 3

Here's an attempt of intuitively justifying it to you. Think about the distance between a point and the origin.

• In $\mathbb{R}^3$ take $P(x,y,z)$ has distance $\sqrt{x^2+y^2+z^2}$
• In $\mathbb{R}^2$ (the $xy$--plane) take $P(x,y)$ has distance $\sqrt{x^2+y^2}$
• In $\mathbb{R}^1$ (the number line ) take $P(x)$ has distance $\sqrt{x^2}=|x|$

So on the number line the distance between any point $P(x)$ and the origin is $|x|$. Now if you want to find the distnce between two points $P(x)$ and $Q(y)$ on the number line first consider their difference $R(x-y)$ it will have the same distance from $0$ as $P$ has from $Q$ so $|b-a|$ is the distance between $a$ and $b$ on the number line. When $b>a$ you can drop the absolute values $|b-a|=b-a$