bounded

Let $X$ be a subset of $ℝ$. We say that $X$ is bounded when there exists a real number $M$ such that for all $x\in X$. When $X$ is an interval, we speak of a bounded interval.

This can be generalized first to ${ℝ}^n$. We say that $X\subseteq {ℝ}^n$ is bounded if there is a real number $M$ such that for all $x\in X$ and $\parallel \cdot \parallel$ is the Euclidean distance between $x$ and $y$.

This condition is equivalent to the statement: There is a real number $T$ such that for all $x,y\in X$.

A further generalization to any metric space $V$ says that $X\subseteq V$ is bounded when there is a real number $M$ such that for all $x,y\in X$, where $d$ is the metric on $V$.