Neighborhoods and Accumulation Points
A neighborhood of a point a is an open interval (a − ε, a + ε) around it, and a point is an accumulation point (limit point) of a set if every neighborhood of it contains infinitely many points of the set. In brief: for the set S = {1/n : n = 1, 2, 3, …}, the point 0 is an accumulation point, since every interval (−ε, ε) contains the numbers 1/n for all n > 1/ε, but 0 is not in S. The points 1, 1/2, 1/3, … are isolated points of S.
Limits describe what happens near a point, and to say “near” precisely, mathematicians use neighborhoods. The concepts of neighborhood, interior point, isolated point and accumulation point are the language in which the definitions of limit and continuity are written. This guide defines them, explains the punctured neighborhood used in limits of functions, introduces open and closed sets, bounded sets and the Bolzano–Weierstrass theorem, with examples and a table of sets and their accumulation points.
What is a neighborhood?
For a real number a and a positive number ε (epsilon), the ε-neighborhood of a is the open interval U(a, ε) = (a − ε, a + ε) = {x : |x − a| < ε}. It consists of all points at distance less than ε from a. The punctured (or deleted) neighborhood excludes the center: (a − ε, a) ∪ (a, a + ε) = {x : 0 < |x − a| < ε}. Punctured neighborhoods are used in the limit of a function at a, where the value at a itself is irrelevant. A neighborhood of +∞ is an interval (M, ∞), and of −∞ an interval (−∞, −M).
Example 1
The 0.5-neighborhood of 3 is (2.5, 3.5). The point 3.2 belongs to it, because |3.2 − 3| = 0.2 < 0.5, but 3.6 does not. The punctured neighborhood of 3 with ε = 0.5 is (2.5, 3) ∪ (3, 3.5).
What kinds of points does a set have?
Let S be a set of real numbers.
- An interior point of S has a neighborhood contained entirely in S.
- A boundary point of S has every neighborhood containing both points of S and points not in S.
- An isolated point of S belongs to S and has a neighborhood with no other points of S.
Example 2
For S = [0, 1): the interior points are those in (0, 1). The boundary points are 0 and 1. The point 0 belongs to S and is a boundary point, but the point 1 does not belong to S. For S = {1, 2, 3}: every point is isolated, with no interior points, and every point is a boundary point.
What is an accumulation point?
A point a (not necessarily in S) is an accumulation point of S if every punctured neighborhood of a contains at least one point of S, equivalently, every neighborhood of a contains infinitely many points of S. Accumulation points are limits of sequences of distinct points of S. A point of S that is not an accumulation point is isolated.
Example 3: 1/n
S = {1, 1/2, 1/3, …}. The only accumulation point is 0. Take any ε > 0, then 1/n ∈ (−ε, ε) for all n > 1/ε, which is infinitely many. Every point 1/n is isolated: for 1/n, the nearest other points are 1/(n + 1) and 1/(n − 1), at positive distance.
Example 4: intervals and rational numbers
Every point of the interval [0, 1] is an accumulation point of (0, 1) and of [0, 1]. Every real number is an accumulation point of the rational numbers ℚ (the rationals are dense), but ℚ has no interior points.
Example 5: finite sets
A finite set has no accumulation points: any neighborhood small enough contains at most one point of the set. The set of integers ℤ has no accumulation points either, since the distance between distinct integers is at least 1.
Example 6: a set with several accumulation points
S = {1/n + (−1)n : n ≥ 1}: the even terms approach 1 and the odd terms approach −1. So the accumulation points are −1 and 1.
What are open and closed sets?
A set is open if each of its points is an interior point (it contains a neighborhood of each of its points): (a, b), (a, ∞) and ℝ are open. A set is closed if it contains all its accumulation points (equivalently, its complement is open): [a, b], ℤ and finite sets are closed. The sets (0, 1] and [0, 1) are neither open nor closed. The empty set and ℝ are both open and closed. A set is bounded if it lies in some interval [−M, M], and a bounded closed set is called compact (in ℝ). This vocabulary is used in the statements about continuous functions and about optimization on closed intervals.
What does the Bolzano–Weierstrass theorem say?
Every bounded infinite set of real numbers has at least one accumulation point. Equivalent form: every bounded sequence has a convergent subsequence. The idea of the proof: bisect the interval containing the set; at least one half contains infinitely many points; repeat, and the nested intervals shrink to a point, which is an accumulation point. The theorem uses the completeness of the real numbers and fails for the rational numbers: the set of rational approximations of √2 is bounded but has no accumulation point in ℚ.
The theorem explains why a bounded sequence such as (−1)n does not converge but has convergent subsequences: its accumulation points (as a sequence) are −1 and 1.
Table of sets and their accumulation points
| Set | Accumulation points | Isolated points | Open or closed? |
|---|---|---|---|
| (0, 1) | all of [0, 1] | none | open |
| [0, 1] | all of [0, 1] | none | closed |
| {1/n : n ≥ 1} | 0 | all points 1/n | neither |
| {0} ∪ {1/n : n ≥ 1} | 0 | all points 1/n | closed |
| ℤ | none | all integers | closed |
| ℚ | all real numbers | none | neither |
| {1, 2, 3} | none | 1, 2, 3 | closed |
Where do these ideas come from?
The concepts of neighborhood and accumulation point were formalised in the nineteenth century by Bernard Bolzano, Karl Weierstrass and Georg Cantor, who studied sets of points while working on the foundations of analysis and Fourier series. The theorem of Bolzano–Weierstrass dates from the 1860s (Bolzano proved a version around 1817). The language of neighborhoods led to topology in the twentieth century, where a neighborhood is defined without a distance.
Where are they used?
The definition of the limit of a function at a point a requires a to be an accumulation point of the domain, so that points of the domain come arbitrarily close to a. In optimization, the existence of maxima and minima on closed bounded sets rests on compactness. In numerical analysis, the convergence of iterative methods is described by neighborhoods of the solution. The same concepts apply to sets in the plane and in space.
Which mistakes are common?
Assuming an accumulation point belongs to the set
The accumulation point 0 of {1/n} is not in the set. An accumulation point can belong to the set or not.
Confusing isolated points with boundary points
Every isolated point is a boundary point, but not conversely: the endpoint 0 of [0, 1] is a boundary point and not isolated.
Thinking that “closed” is the opposite of “open”
Some sets, such as [0, 1), are neither, and ℝ is both.
Using a neighborhood with ε = 0 or negative
The radius ε must be positive. The definitions of limits require that the statement holds for every ε > 0.