A real number x is an accumulation point of a real-numbered set if every neighborhood of x contains infinitely many elements of the set [1]. To put this another way, point x is an accumulation point if, for any arbitrarily close neighborhood of x you can find elements of the set that are different from x.
To be an accumulation point x, there have to be points in the set arbitrarily close to x other than x itself. You can think of x as being similar to a limit of a sequence of points of S that can’t just be the constant sequence x [2].
An accumulation point is sometimes called a limit point [3]; This stems from the idea is that x is a limit of a non-constant sequence in the set. A point x can be approximated by other points in the set, much in the same way a limit can be found by studying the behavior of a function as it tends towards a certain point.
An accumulation point is a useful way to determine whether or not a set is a closed set; a set is closed if it contains all of its limit points.
Accumulation Point Examples
A couple of examples [2]:
| Set | Accumulation points |
| [−1, 16) | [−1, 16] |
| [−1, 16) ∪ (16, 20] | [−1, 20] |
The Bolzano–Weierstrass theorem implies that every bounded infinite set has an accumulation point.
Examples of Sets without an Accumulation Point
- Finite sets. The singleton set {5} is one example. No matter which neighborhood of 5 you consider, you won’t be able to find any elements in the set that are different from 5.
- The set of natural numbers ℕ. There is no accumulation point because a neighborhood with width ⅓ contains at most on natural number [5].
References
[1] Smith, W. (2018). Solutions. Retrieved May 28, 2021 from: http://www.math.hawaii.edu/~wayne/331/HW3.pdf
[2] Fogel, M. The Bolzano-Weierstrass Theorem. Retrieved May 28, 2021 from: http://staff.imsa.edu/~fogel/Analysis/PDF/15%20Bolzano-Weierstrass%20Theorem.pdf
[3] Pelayo, R. Math 431 — Real Analysis I. Retrieved May 28, 2021 from: http://www2.hawaii.edu/~robertop/Courses/Math_431/Handouts/Quest_1_sols.pdf
[4] Berger, L. (2012). Comments on boundary points and accumulation points. Retrieved May 28, 2021 from: https://www.math.stonybrook.edu/~lbrgr/513BoundaryPoints2012.pdf
[5] Boas, Dr. (2011). Advanced Calculus I. Retrieved May 28, 2021 from: https://www.math.tamu.edu/~boas/courses/409-2017a/solution1from2011.pdf