Appendix BReal Analysis

B.1 Set Theory

Here we give some basic definitions from set theory and topology.

Let be a vector in . An (open) neighborhood of , also called an open ball of radius at , is the set
$upper B left-parenthesis x 0 comma epsilon right-parenthesis equals StartSet x element-of double-struck upper R Superscript n Baseline vertical-bar vertical-bar vertical-bar x minus x 0 vertical-bar vertical-bar less-than epsilon EndSet period$

A closed neighborhood of is the set

$ModifyingAbove upper B With bar left-parenthesis x 0 comma epsilon right-parenthesis equals StartSet x element-of double-struck upper R Superscript n Baseline vertical-bar vertical-bar vertical-bar x minus x 0 vertical-bar vertical-bar less-than-or-equal-to epsilon EndSet period$
A set is an open set if, for each , there is a neighborhood such that .
A set is a closed set if and only if the complement of , , meaning the ...