A system of real numbers, or of the points on a straight line corresponding to them, defined in any way whatever, is called an aggregate or set of numbers or points. The set might consist, for example, of all the positive integers, or of all the rational points.

It is most convenient here to use the language of geometry.1 Suppose then that we are given a set of points, which we will denote by $$S$$. Take any point $$\xi$$, which may or may not belong to $$S$$. Then there are two possibilities. Either (i) it is possible to choose a positive number $$\delta$$ so that the interval $${[\xi – \delta, \xi + \delta]}$$ does not contain any point of $$S$$, other than $$\xi$$ itself,2 or (ii) this is not possible.

Suppose, for example, that $$S$$ consists of the points corresponding to all the positive integers. If $$\xi$$ is itself a positive integer, we can take $$\delta$$ to be any number less than $$1$$, and (i) will be true; or, if $$\xi$$ is halfway between two positive integers, we can take $$\delta$$ to be any number less than $$\frac{1}{2}$$. On the other hand, if $$S$$ consists of all the rational points, then, whatever the value of $$\xi$$, (ii) is true; for any interval whatever contains an infinity of rational points.

Let us suppose that (ii) is true. Then any interval $${[\xi – \delta, \xi + \delta]}$$, however small its length, contains at least one point $$\xi_{1}$$ which belongs to $$S$$ and does not coincide with $$\xi$$; and this whether $$\xi$$ itself be a member of $$S$$ or not. In this case we shall say that $$\xi$$ is a point of accumulation of $$S$$. It is easy to see that the interval $${[\xi – \delta, \xi + \delta]}$$ must contain, not merely one, but infinitely many points of $$S$$. For, when we have determined $$\xi_{1}$$, we can take an interval $${[\xi – \delta_{1}, \xi + \delta_{1}]}$$ surrounding $$\xi$$ but not reaching as far as $$\xi_{1}$$. But this interval also must contain a point, say $$\xi_{2}$$, which is a member of $$S$$ and does not coincide with $$\xi$$. Obviously we may repeat this argument, with $$\xi_{2}$$ in the place of $$\xi_{1}$$; and so on indefinitely. In this way we can determine as many points $\xi_{1},\quad \xi_{2},\quad \xi_{3},\ \dots$ as we please, all belonging to $$S$$, and all lying inside the interval $${[\xi – \delta, \xi + \delta]}$$.

A point of accumulation of $$S$$ may or may not be itself a point of $$S$$. The examples which follow illustrate the various possibilities.

Example IX

1. If $$S$$ consists of the points corresponding to the positive integers, or all the integers, there are no points of accumulation.

2. If $$S$$ consists of all the rational points, every point of the line is a point of accumulation.

3. If $$S$$ consists of the points $$1$$, $$\frac{1}{2}$$, $$\frac{1}{3}, \dots$$, there is one point of accumulation, viz. the origin.

4. If $$S$$ consists of all the positive rational points, the points of accumulation are the origin and all positive points of the line.

1. The reader will hardly require to be reminded that this course is adopted solely for reasons of linguistic convenience.↩︎
2. This clause is of course unnecessary if $$\xi$$ does not itself belong to $$S$$.↩︎