Katie Rogier, Jennifer Medeiros, and Julie Mello
Definition 5.22.
Let A ⊂ R be a subset of R. Then x ∈ R is:
(1) an interior point of A if there exists δ>0 such that A ⊃ (x−δ,x+δ);
(2) an isolated point of A if x ∈ A and there exists δ>0 such that x is the only point in A that belongs to the interval (x − δ, x + δ);
(3) a boundary point of A if for every δ > 0 the interval (x − δ,x + δ) contains points in A and points not in A;
(4) an accumulation point of A if for every δ > 0 the interval (x−δ, x+δ) contains a point in A that is distinct from x.
When the set A is understood from the context, we refer, for example, to an “interior point.”
Interior and isolated points of a set belong to the set, whereas boundary and accumulation points may or may not belong to the set. In the definition of a boundary point x, we allow the possibility that x itself is a point in A belonging to (x − δ, x + δ), but in the definition of an accumulation point, we consider only points in A belonging to (x − δ, x + δ) that are distinct from x. Thus an isolated point is a boundary point, but it isn’t an accumulation point. Accumulation points are also called cluster points or limit points.
We illustrate these definitions with a number of examples.
Example 5.23.
Let I = (a, b) be an open interval and J = [a, b] a closed interval. Then the set of interior points of I or J is (a,b), and the set of boundary points consists of the two endpoints {a,b}. The set of accumulation points of I or J is the closed interval [a,b] and I, J have no isolated points. Thus, I, J have the same interior, isolated, boundary and accumulation points, but J contains its boundary points and all of its accumulation points, while I does not.
Example 5.24.
Let a < c < b and suppose that A = (a, c) ∪ (c, b)
is an open interval punctured at c. Then the set of interior points is A, the set of boundary points is {a, b, c}, the set of accumulation points is the closed interval [a, b], and there are no isolated points.
Example 5.25.
Let A=n:n∈N.
Then every point of A is an isolated point, since a sufficiently small interval about 1/n doesn’t contain 1/m for any integer m ̸= n, and A has no interior points. The set of boundary points of A is A ∪ {0}. The point 0 ∈/ A is the only accumulation point of A, since every open interval about 0 contains 1/n for sufficiently large n.
Example 5.26.
The set N of natural numbers has no interior or accumulation points. Every point of N is both a boundary point and an isolated point.
Example 5.27.
The set Q of rational numbers has no interior or isolated points, and every real number is both a boundary and accumulation point of Q.
Example 5.28.
The Cantor set C defined in Section 5.5 below has no interior points and no isolated points. The set of accumulation points and the set of boundary points of C is equal to C.
The following proposition gives a sequential definition of an accumulation point.
Proposition 5.29.
A point x ∈ R is an accumulation point of A ⊂ R if and onl