Answer to Question #319104 in Math for Silling

Question

Determine the set of interior points, exterior points,

accumulation points, isolated points, and boundary

points of the set E = {x : x

2 ≥ 2}.

Answer 1

$E=\left\{ x:x^2\geqslant 2 \right\} =\left( -\infty ,-\sqrt{2} \right] \cup \left[ \sqrt{2},+\infty \right)$

Interior points are $\left( -\infty ,-\sqrt{2} \right) \cup \left( \sqrt{2},+\infty \right)$ since, for each of these points there is an open interval inside E but for $\pm \sqrt{2}$ there is not.

Exterior points are $\left( -\sqrt{2},\sqrt{2} \right)$ since these are all the points outside E, and for each of these points there is an open interval outside E.

Accumulation points are all points of E, since for each of them every open interval with center in this point contains other point of E. There are no other accumulation points since other points of the line are exterior.

No isolated points since all points of E are accumulation points

Boundary points are neither interior nor exterior, thus $\left\{ -\sqrt{2},\sqrt{2} \right\}$