 #1
Eclair_de_XII
 1,067
 90
 TL;DR Summary

On the real numbers, a set ##D## is bounded iff there exists a positive ##M## such that for all ##y\in D##, ##y\leq M##.
On the real numbers, a set ##D## is unbounded iff for all positive ##M##, there is a ##y\in## such that ##y>M##.
I read in my textbook Calculus on Manifolds by Spivak that a set ##A\subset \mathbb{R}^n## is bounded iff there is a closed nrectangle ##D## such that ##A\subset D##. It should be plain that if I wanted to define unboundedness, I should just say something along the lines of: "A set ##A\subset \mathbb{R}^n##is unbounded iff for all nrectangle ##D##, ##A## is not contained in ##D##.D
So my question is if the following definition of unboundedness is valid: "A set ##A## is unbounded if for all nrectangles ##D##, there is a ##y\in A## such that there is an open set containing ##y## that is a subset of ##\mathbb{R}^n \cap D^c##".
So my question is if the following definition of unboundedness is valid: "A set ##A## is unbounded if for all nrectangles ##D##, there is a ##y\in A## such that there is an open set containing ##y## that is a subset of ##\mathbb{R}^n \cap D^c##".