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This lesson covers a fairly advanced topic: the definition of the boundary points of a set.

You will learn an intuitive way to visualize the boundary points and the precise definition of boundary point that is used in mathematics.

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Introduction: An Intuitive Way to Think about Boundary Points

When you think of the word boundary, what comes to mind? Maybe the clearest real-world examples are the state lines as you cross from one state to the next. Well, if you consider all of the land in Georgia as the points belonging to the set called Georgia, then the boundary points of that set are exactly those points on the state lines, where Georgia transitions to Alabama or to South Carolina or Florida, etc. Intuitively, a boundary point of a set is any point on the edge, or border, separating the interior from the exterior of the set.

This shows Georgia in red. Every point on the border between Georgia and another state (or the Atlantic ocean) is a boundary point of Georgia as a set.
Open r-neighborhood

I like to think of P being the location of a pole in my backyard, and I’ve tied a dog to this pole. The dog can visit any part of my yard that is no more than the length of the rope (r) away from P — that region that the dog can romp around in is something like an r-neighborhood around P.

The Precise Definition of Boundary Point

Given a set S and a point P (which may not necessarily be in S itself), then P is a boundary point of S if and only if every neighborhood of P has at least a point in common with S and a point not in S.For example, in the picture below, if the bluish-green area represents a set S, then the set of boundary points of S form the darker blue outlines.

A set with boundary points highlighted

So what does this really mean? Well think back to the state lines of your home state. I live in Georgia, and if I travel south far enough, I’ll enter Florida.

Suppose I drive right up to the Florida border and get out of my car and put a pole in the ground on the state line. I tie my dog (who travels with me everywhere) to this pole. Then, no matter how short the rope is, the dog can visit points in both Georgia and not in Georgia (in Florida). That means the state line really does represent a boundary of the set of points of Georgia!

Example

Suppose S is the set of all points in a closed disk of radius 3 centered at (1, -2), as shown in the figure. What are the boundary points?

Circle example

Some boundary points of S include: (1,1), (4,-2), etc. In fact, the boundary of S is just the set of points on the circumference of the disk.

Lesson Summary

The points of the boundary of a set are, intuitively speaking, those points on the edge of S, separating the interior from the exterior. More precisely, a point P is a boundary point of a set S if every neighborhood of P contains at least one point in S and one point not in S.

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