Boundary Points

In today's blog, I define boundary points and show their relationship to open and closed sets.

Definition 1: Boundary Point

A point x is a boundary point of a set X if for all ε greater than 0, the interval (x - ε, x + ε) contains a point in X and a point in X'.

Lemma 1: A set is open when it contains none of its boundary points and it is closed when it contains all of its boundary points.

Proof:

(1) Let a,b be the boundary points for a set S of real numbers that are not part of S where a is the lower bound and b is the upper bound.

(2) If a,b are not included in S, then we have S = { x : x is greater than a and less than b } which means that x is an open set. [See Lemma 5, here]

(3) If a,b are included in S, then we have S = { x : a ≤ x ≤ b } which means that x is a closed set. [See Lemma 7, here]

QED

Lemma 2: Every real number is a boundary point of the set of rational numbers Q.

Proof:

(1) A boundary point b by definition is a point where for any positive number ε, { b - ε , b + ε } contains both an element in Q and an element in Q'.

(2) So all we need to show that { b - ε, b + ε } contains both a rational number and an irrational number.

(3) We know that this is the case based on the properties of the set of rational numbers and the properties of the set of real numbers.

We can find know that there is at least one rational number in { b - ε, b + ε } [See Corollary 2.1, here]

By the nature of the continuity of real numbers, there exists an irrational number r such that abs(b - r) is less than ε. [See here for a review of the properties of irrational numbers]

QED

References

• Charles R. MacCluer, Honors Calculus, Princeton University Press, 2006
• Bob Gardner, Introductory Level Analysis: Synthesizing R, Rⁿ, Metric Spaces and Topological Spaces.