Bounded Set

Definition

A set is bounded if all of its points are within a certain distance of each other. The notion of boundedness is makes sense with some metric.

Consider an Partially Ordered Set $X$ and a subset $Y\subset X$. The set $Y$ is bounded if it has both upper and lower bounds $\exists x_0,x_1\in X,\text{s.t.}(\forall y\in Y,y\le x_0)\text{and}(\forall y\in Y,y\ge x_1)$ where $x_0,x_1$ is the upper and lower bound of the set $Y$ respectively.

Definition of Bounded Set on a Metric Space

Consider a Metric Space $(X,\mathcal{T},d)$ and a subset $Y\subset X$. The set $Y$ is bounded if $\exists r>0\text{s.t.}\forall y_0,y_1\in Y,d(y_0,y_1)<r$