Bounded Set in a Metric Space¶

Definition¶

Given a metric space $(X,d_{X})\text{,}$ a set $E\subseteq X$ is termed bounded if there exists $M>0$ such that the ball $B_{(X,d_{X})} (x,M)$ contains $E\text{.}$ Here, $x$ can be any point in $X$ (usually chosen to be the origin or a particular point in $E$).

Interpretation¶

A bounded set is one that fits entirely inside some large enough ball, so that no two of its points are arbitrarily far apart. It extends the notion of a bounded set of real numbers to any metric space.

The whole set $E$ lies inside the dashed ball of radius $M$ around $x_0$, so $E\subseteq B(x_0,M)$$B(x_0,M)$. The ball is only a witness: it need not be small or centred in $E$, and any larger radius works as well.

The choice of centre does not matter. If $E\subseteq B(x,M)$ and $y$ is any other point, then for every $z\in E$ the triangle inequality gives $d(z,y)\le d(z,x)+d(x,y)<M+d(x,y)$$d(x,y)$, so $E\subseteq B(y,M+d(x,y))$$d(x,y))$; only the radius changes. Equivalently, as in the Key Results, a nonempty set is bounded if and only if its diameter $\sup_{x,y\in E}d(x,y)$ is finite.

In $\mathbb{R}^n$ with the Euclidean distance, a set is bounded if and only if all coordinates of its points are bounded. Boundedness is one of the two conditions in the Heine-Borel theorem, and a bounded sequence in $\mathbb{R}^n$ has a convergent subsequence by the Bolzano-Weierstrass theorem.

Boundedness depends on the metric, not only on the set. Under the discrete metric every set, including $\mathbb{R}$ itself, is bounded, because all distances are at most $1$. For this reason "closed and bounded" implies compact in $\mathbb{R}^n$ but not in a general metric space.

Key Results¶

• A nonempty set is bounded if and only if $\displaystyle \sup_{x,y\in E} d_{X}(x,y)$ is finite.
• Any subset of a bounded set is bounded.

Examples¶

• A unit sphere $\{(x,y,z)\in\mathbb{R}^3\colon x^2+y^2+z^2=1\}$ is bounded.
• A parabola curve $\{(x,y)\in\mathbb{R}^2\colon y=x^2\}$ is unbounded.

Prerequisites¶

• Open Metric Ball

Related Concepts¶

• Bounded Set

Sources¶

• Math III: Analysis · Lecture 9

Symbols: element of, real numbers, subset

Last updated: 20 October 2026 03:02