What Is the Difference Between Bounded, Unbounded, Open, and Closed Sets?

The words open, closed, bounded, and unbounded describe two different features of a set. Open and closed concern the set’s boundary and nearby points. Bounded and unbounded concern how far the set extends. These properties combine as follows: every set is either bounded or unbounded, and a set can be open, closed, both, or neither.

The definitions depend on the space and distance you are using. Unless stated otherwise, the examples below are subsets of the real line $\mathbb{R}$ with its usual distance.

What bounded and unbounded mean

A set $S\subseteq\mathbb{R}$ is bounded if its elements all lie within some finite distance of the origin. In symbols, there is a number $M<\infty$ such that $|x|\le M$ for every $x\in S$. Equivalently, the set fits inside some finite interval. A set is unbounded if no such $M$ exists.

For example, $(0,1)$ is bounded because every element lies between $-1$ and $1$. The set $[0,\infty)$ is unbounded: however large a finite number $M$ you choose, the set contains numbers larger than $M$.

In a metric space, the same idea is expressed using distance: a set is bounded if all its points fit inside some ball of finite radius. Boundedness is about size, not whether endpoints or boundary points are included.

What open and closed mean

A set $S\subseteq\mathbb{R}$ is open if, around each point $x\in S$, you can find a small interval that remains entirely inside $S$. That is, for every $x\in S$, there is an $\varepsilon>0$ such that

$$(x-\varepsilon,x+\varepsilon)\subseteq S.$$

A set is closed if it contains all of its limit points. An equivalent definition is that its complement in the ambient space is open. For example, $[0,1]$ is closed in $\mathbb{R}$ because it includes its endpoints, which are limit points of the interval.

The definitions are not simply “does it include its endpoints?” That shortcut can help with intervals, but it does not describe open or closed sets in general. The neighborhood and complement definitions work for arbitrary sets.

A worked classification

Consider four subsets of $\mathbb{R}$:

Set	Open?	Closed?	Bounded?
$(0,1)$	Yes	No	Yes
$[0,1]$	No	Yes	Yes
$(0,1]$	No	No	Yes
$[0,\infty)$	No	Yes	No

Here is why the classifications differ:

• $(0,1)$ is open because each of its points has a small interval around it that stays between $0$ and $1$. It is not closed because it omits its limit points $0$ and $1$. It is bounded because all its points lie between $0$ and $1$.
• $[0,1]$ is closed because it contains its limit points, including $0$ and $1$. It is not open in $\mathbb{R}$: no interval around $0$ fits inside $[0,1]$, since such an interval includes negative numbers. It is bounded.
• $(0,1]$ is neither open nor closed. It is not open because a neighborhood around $1$ extends beyond $1$. It is not closed because it omits the limit point $0$. It is nevertheless bounded.
• $[0,\infty)$ is closed because its complement, $(-\infty,0)$, is open. It is unbounded because it has no finite upper bound.

The empty set $\varnothing$ and the whole real line $\mathbb{R}$ are both open and closed. In the usual topology on $\mathbb{R}$, these are the only sets that are both open and closed. The whole line is unbounded, while the empty set is bounded under the usual definition.

Why the two pairs are independent

Boundedness does not decide whether a set is open or closed. The examples $(0,1)$ and $[0,1]$ are both bounded, but one is open and the other closed. Likewise, the unbounded set $[0,\infty)$ is closed, while $\mathbb{R}$ is both open and closed.

In Euclidean space $\mathbb{R}^n$, the Heine–Borel theorem says that a set is compact exactly when it is both closed and bounded. This is a useful connection, but it does not make “closed” and “bounded” synonyms. The equivalence between compactness and being closed and bounded does not hold in every metric space.

Also, openness and closedness are relative to the ambient space. For instance, $[0,1]$ is not open as a subset of $\mathbb{R}$, but it is open if you regard $[0,1]$ itself as the entire space. When a problem specifies a space or topology, use that setting for the definitions.

Common mistakes

• Treating open and closed as opposites. A set can be both, as $\mathbb{R}$ is, or neither, as $(0,1]$ is.
• Assuming bounded means closed. The open interval $(0,1)$ is bounded but not closed.
• Assuming closed means bounded. The ray $[0,\infty)$ is closed but unbounded.
• Checking only endpoints. Endpoint tests are convenient for intervals, but the general tests are neighborhoods for openness and limit points (or an open complement) for closedness.
• Ignoring the ambient space. A set’s classification can change when it is considered as a subset of a different space.