5.4.1 Introduction

There are several areas of topology, such as graph theory and network topology, geometric topology, algebraic topology, differential topology, combinatorial topology, and point‐set (or general) topology. Some areas of topology are sufficiently diverse that practicing researchers in one area may have only a passing knowledge of other areas. While graph theory is interested in connections between objects, and geometric topology was interested in shapes of objects, point‐set topology is about “closeness.” By closeness, we mean closeness of numbers, points, functions, matrices, operators, and other mathematical objects. The interest in closeness lies in the fact it gives rise to limits, infinite series, convergence, continuity, and mathematical concepts associated with calculus, differential equations, real and complex analysis, as well as areas of science and engineering.

Point‐set topology can be studied in any dimension, even infinite, but in this section, we restrict our attention to the one‐dimensional real line, mostly because it is easier to visualize than in two and three dimensions, and a whole lot simpler than dimensions larger than three.¹ The basic concepts of point‐set topology begin with open intervals and the concept of neighborhoods.

This brings us to the unifying concept of this section, and in much of point‐set topology, the open set.

In the definition of an open set, when we say “there exists a δ greater than zero,” we normally are thinking of a small δ, not a large one. In plain language, a set is open if you can “wiggle” any point in the set around, and if you do not wiggle it around too much it will still be in the set. Still another way of thinking about open sets is that every point in an open set is surrounded by other points in the set, i.e. every point in the set is insulated from the outside.

In other words, a set is not open if there exists at least one point a ∈ A right on the boundary of A and not insulated from the outside by other points of A. The interval (a, b] is not open since the point x = b cannot be “wiggled” any amount without finding itself outside (a, b].

The most important properties of open sets relate to their union and intersection.

5.4.2 Interior, Exterior, and Boundary of a Set

Three important concepts of topology are the concepts of interior, exterior, and boundary of a set.⁵

5.4.3 Interiors, Boundaries, and Exteriors of Common Sets

Table 5.5 shows the interiors, boundaries, and exteriors of some common sets. Note that the interior and exterior of a set are always open. Also, note that the exterior is the complement of the union of the interior and boundary, and is always closed. Note too that the union of the interior, boundary, and exterior of a set is everything. In other words, the real numbers ℝ.

A ⊆ ℝ	Int(A) ⊆ A	Bdy(A)	Ext(A)
Ø	Ø	Ø	ℝ
(a, b)	(a, b)	{a, b}	(− ∞ , a) ∪ (b, ∞)
[a, b]	(a, b)	{a, b}	(− ∞ , a) ∪ (b, ∞)
(a, b]	(a, b)	{a, b}	(− ∞ , a) ∪ (b, ∞)
{a}	Ø	{a}	(− ∞ , a) ∪ (a, ∞)
{a, b, c}	Ø	{a, b, c}	ℝ − {a, b, c}
	Ø
(0, 1) ∪ {2}	(0, 1)	{0, 1, 2}	(− ∞ , 0) ∪ (1, 2) ∪ (2, ∞)
(−1, 0) ∪ (0, 1)	(−1, 0) ∪ (0.1)	{−1, 0, 1}	(− ∞ , − 1) ∪ (1, ∞)
ℕ	Ø	ℕ	ℝ − ℕ
ℤ	Ø	ℤ	ℝ − ℤ
ℚ	Ø	ℝ	Ø
ℝ	ℝ	Ø	Ø

5.4.4 Limit Points

The concept of a limit is fundamental in calculus and analysis. The reader will recall that the two fundamental operations of the calculus, the derivative and integral, are both limits. We can thank the French mathematician Augustin‐Louis Cauchy (1789–1857) and German mathematician Karl Weierstrass (1815–1897) for providing rigorous definition of the limit, the so‐called epsilon, delta (ε, δ) definition, which allows mathematicians to reason with precision the ideas of calculus and avoid the imprecise reasoning of the past.

5.4.5 Closed Sets Contain Their Limit Points

We have seen that a good way to determine if a set is closed is to show that its complement is open. Although this indirect procedure is useful, there is a direct way to determine if a set is closed. The following theorem makes this precise.

Table 5.6 gives the limit points of some common sets of real numbers.

A ⊆ ℝ	Limit points(A)	Limit points(A) ⊆ A?
Ø	Ø	Closed
(a, b)	[a, b]	Open
[a, b]	[a, b]	Closed
(a, b]	[a, b]	Neither open or closed
{a}	Ø	Closed
{a, b, c}	Ø	Closed
	{0}	Neither open or closed
(0, 1) ∪ {2}	[0, 1]	Neither open or closed
[0, 1] ∪ {2}	[0, 1]	Closed
ℕ	Ø	Closed
ℤ	Ø	Closed
ℚ	ℝ	Neither open or closed
ℝ	ℝ	Closed

5.4.6 Topological Spaces

We saw in Chapter 4 how relations (ordering, equivalence, function, …) can be defined on a set so that the elements of the set can be compared, ordered, and classified. We now introduce a different structure on a set, a topology, which places a general “distance” structure on a set, allowing one to make precise concepts like closeness, limits, and all those concepts used in calculus, like continuous functions, derivatives, and integrals.

We now show how open sets are the foundation of a topological structure on a set, in our case the real numbers ℝ.

5.4.7 Calculus Without Topology Is No Calculus

Recall that functions f : ℝ → ℝ are the basis of many areas of mathematics, like calculus, which the reader no doubt has some familiarity. If there is no topological structure on ℝ, there are no limits, continuity, differentiable and integral functions, and so on. When a student of calculus thinks about the real numbers, the student most likely is thinking about the topological structure of the real numbers and the distance between points, and so on.⁷

Topology allows one to make precise many of the important concepts of mathematics, one of the most important being continuous functions. We now give the topological definition of continuity.