Open and Closed Sets in R

Open and closed sets form the foundation of topology, the study of continuous deformations and limits. In $\mathbb{R}$ , open sets are intervals without endpoints, while closed sets contain all their limit points. These concepts generalize to arbitrary metric spaces and topological spaces, making them central to modern analysis.

Definitions

Definition3.1Open set

A set $U \subseteq \mathbb{R}$ is open if for every $x \in U$ , there exists $\epsilon > 0$ such that the interval $(x - \epsilon, x + \epsilon) \subseteq U$ .

Equivalently, $U$ is open if every point of $U$ is an interior point: a point $x$ is interior to $U$ if some neighborhood of $x$ is contained in $U$ .

Definition3.2Closed set

A set $F \subseteq \mathbb{R}$ is closed if its complement $\mathbb{R} \setminus F$ is open.

Equivalently, $F$ is closed if it contains all its limit points: whenever a sequence $(x_n)$ in $F$ converges to some $L \in \mathbb{R}$ , we have $L \in F$ .

ExampleOpen intervals are open

The interval $(a, b) = \{x \in \mathbb{R} \mid a < x < b\}$ is open. For any $x \in (a, b)$ , let $\epsilon = \min(x - a, b - x) > 0$ . Then $(x - \epsilon, x + \epsilon) \subseteq (a, b)$ .

ExampleClosed intervals are closed

The interval $[a, b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}$ is closed. Its complement is $(-\infty, a) \cup (b, \infty)$ , which is open (a union of two open sets). Alternatively, if $x_n \in [a, b]$ and $x_n \to L$ , then $a \leq x_n \leq b$ for all $n$ , so taking limits, $a \leq L \leq b$ , hence $L \in [a, b]$ .

ExampleHalf-open intervals are neither open nor closed

The interval $[a, b) = \{x \in \mathbb{R} \mid a \leq x < b\}$ is neither open nor closed. It is not open because $a \in [a, b)$ but no interval $(a - \epsilon, a + \epsilon)$ is contained in $[a, b)$ . It is not closed because the complement $(-\infty, a) \cup [b, \infty)$ is not open (no neighborhood of $b$ is contained in the complement).

ExampleEmpty set and R

Both $\varnothing$ and $\mathbb{R}$ are both open and closed (clopen). This is consistent: $\mathbb{R} \setminus \varnothing = \mathbb{R}$ is open, so $\varnothing$ is closed. Similarly, $\mathbb{R} \setminus \mathbb{R} = \varnothing$ is open (vacuously), so $\mathbb{R}$ is closed.

Properties of open and closed sets

Theorem3.1Properties of open sets

The union of any collection of open sets is open.
The intersection of finitely many open sets is open.
$\varnothing$ and $\mathbb{R}$ are open.

Theorem3.2Properties of closed sets

The intersection of any collection of closed sets is closed.
The union of finitely many closed sets is closed.
$\varnothing$ and $\mathbb{R}$ are closed.

RemarkDe Morgan's laws

These properties are dual by De Morgan's laws: the complement of a union is the intersection of complements, and vice versa. Property 2 for open sets requires finite intersection (infinite intersections can be non-open: $\bigcap_{n=1}^\infty (-1/n, 1/n) = \{0\}$ is not open).

ExampleInfinite intersection of open sets

Let $U_n = (-1/n, 1/n)$ for $n \geq 1$ . Each $U_n$ is open, but

$\bigcap_{n=1}^\infty U_n = \{0\}$

is not open (no neighborhood of $0$ is contained in $\{0\}$ ). Thus infinite intersections of open sets need not be open.

ExampleInfinite union of closed sets

Let $F_n = [1/n, 1]$ for $n \geq 1$ . Each $F_n$ is closed, but

$\bigcup_{n=1}^\infty F_n = (0, 1]$

is not closed (its complement $(-\infty, 0] \cup (1, \infty)$ is not open). Thus infinite unions of closed sets need not be closed.

Limit points and closure

Definition3.3Limit point

A point $x \in \mathbb{R}$ is a limit point (or accumulation point) of a set $E \subseteq \mathbb{R}$ if every neighborhood of $x$ contains a point of $E$ distinct from $x$ . Equivalently, there exists a sequence $(x_n)$ in $E \setminus \{x\}$ with $x_n \to x$ .

Definition3.4Closure

The closure of $E$ , denoted $\overline{E}$ or $\text{cl}(E)$ , is the set $E$ together with all its limit points:

$\overline{E} = E \cup \{x \mid x \text{ is a limit point of } E\}.$

Equivalently, $\overline{E}$ is the smallest closed set containing $E$ (the intersection of all closed sets containing $E$ ).

ExampleClosure of Q

The closure of $\mathbb{Q}$ in $\mathbb{R}$ is $\overline{\mathbb{Q}} = \mathbb{R}$ . Every real number is a limit of a sequence of rationals (by density), so every real is a limit point of $\mathbb{Q}$ .

ExampleClosure of (a, b)

The closure of $(a, b)$ is $\overline{(a, b)} = [a, b]$ . The limit points of $(a, b)$ are all points in $[a, b]$ (including the endpoints $a$ and $b$ ).

ExampleIsolated points

The set $E = \{1, 1/2, 1/3, 1/4, \ldots\}$ has $0$ as its only limit point. Thus $\overline{E} = E \cup \{0\} = \{0, 1, 1/2, 1/3, \ldots\}$ . The points $1, 1/2, 1/3, \ldots$ are isolated (not limit points of $E$ ).

Interior, boundary, and exterior

Definition3.5Interior, boundary, exterior

Let $E \subseteq \mathbb{R}$ .

The interior of $E$ is $\text{int}(E) = \{x \in E \mid \exists \epsilon > 0, (x - \epsilon, x + \epsilon) \subseteq E\}$ (the largest open set contained in $E$ ).
The boundary of $E$ is $\partial E = \overline{E} \setminus \text{int}(E)$ (points in the closure but not in the interior).
The exterior of $E$ is $\text{ext}(E) = \text{int}(\mathbb{R} \setminus E)$ (interior of the complement).

ExampleBoundaries

$\partial(a, b) = \{a, b\}$ (endpoints).
$\partial[a, b] = \{a, b\}$ (same boundary as $(a, b)$ ).
$\partial\mathbb{Q} = \mathbb{R}$ (every real is a boundary point of $\mathbb{Q}$ , since $\mathbb{Q}$ is dense).
$\partial\{0\} = \{0\}$ (a single point is its own boundary).

Summary

Open and closed sets are fundamental in topology:

Open sets: intervals without endpoints, unions of open intervals.
Closed sets: contain all limit points, complements of open sets.
Arbitrary unions of open sets are open; finite intersections are open.
Arbitrary intersections of closed sets are closed; finite unions are closed.
Closure, interior, and boundary capture different aspects of a set's topology.

See Compactness and Heine-Borel for applications to analysis.