Vector Calculus: An Introduction

You are certainly familiar with the notion of open and closed intervals in $\mathbb{R}\text{.}$ In this section we want to generalise these notions to subsets of ${\mathbb{R}}^N\text{.}$ We call an interval $I\subset \mathbb{R}$ open, if its endpoints do not belong to $I\text{.}$ Hence, if we pick an arbitrary point in $x_0\in I$ and move a little bit to the left or to the right we stay in the interval. Mathematically we can express this by saying that for every $x_0\in I$ there exists a small interval $(x_0-\epsilon ,x_0+\epsilon )$ which is entirely contained in $I\text{.}$ If $I$ is an arbitrary interval we call such a point $x_0\in I$ an interior point of $I\text{.}$ The same ideas can be applied to subsets of ${\mathbb{R}}^N\text{.}$ A point ${\mathit{x}}_0$ in $U\subset {\mathbb{R}}^N$ is called an interior point if we can move a little bit in every direction without immediately leaving $U\text{.}$ Mathematically we can express this by requiring that a small ball about ${\mathit{x}}_{\mathbf{0}}$ be entirely contained in the set $U\text{.}$

If $N=1$ then $B_r(x_0)$ is simply the interval $(x_0-r,x_0+r)\text{.}$ If $N=2$ then $B_r({\mathit{x}}_0)$ is an (open) disc centred at ${\mathit{x}}_0$ with radius $r\text{,}$ and for $N=3$ it is really a ball. We could distinguish between these different cases and expressions, but it is more convenient to call $B_r({\mathit{x}}_0)$ simply a “ball,” no matter what the ‘dimension’ of the space is.

As motivated before we make the following definition.

Certain sets have only interior points. They are called open sets.

If $I$ is an interval, its endpoints may or may not belong to it. In any case they play a special role. They are the points where we can ‘enter’ or ‘exit’ the interval with an arbitrarily small move. The set of all endpoints of an interval we also call the boundary of the interval. For subsets of ${\mathbb{R}}^N$ the boundary is defined the same way, namely as the set of points, where we can ‘enter’ or ‘exit’ the set with an arbitrarily small move. Here is a precise mathematical definition.

In Figure 3.11 interior and boundary points of a set in ${\mathbb{R}}^2$ are shown, together with a ball centred at them as required in the definition. Note that, as we get close to the boundary of the set the balls about an interior point might be very small.

Moreover, as with the endpoints of an interval, the boundary of a set may or may not belong to the set. Sometimes only part of the boundary of a set belongs to the set, as with a half open interval.

Sometimes it is useful to look at the ‘closure’ of a set.

We want to give now some examples illustrating the above definitions.