## Fall 2004 #1

Problem Statement: (a) Given a set $S\subseteq \mathbb{R}$ define what it means for a point $p$ to be an interior point of $S$. (b) For $X\subseteq\mathbb{R}$ let $S^{0}=\{p\in\mathbb{R}|p \textup{ is an interior point of } S\}$. Show that $S^{0}$ is open in $S$.

(a)Definition: A point $p$ is an interior point of $S\subseteq\mathbb{R}$ if there exists a real number $c>0$ such that $(p-c,p+c)\subseteq S$.

(b) Proof: We wish to show that every point of $S^{0}$ is an interior point of $S^{0}$. Let $p\in S^{0}$, then $p$ is an interior point of $S$ and so there exists some $c>0$ such that $(p-c,p+c)\subseteq S$.

Now pick $q\in (p-c,p+c)$ and define $\delta>0$ to be the largest possible value such that $(q-\delta,q+\delta)\subseteq (p-c,p+c)$. We know such a $\delta$ exists since $(p-c,p+c)$ is an open set. Then it follows that $(q-\delta,q+\delta)\subseteq (p-c,p+c)\subseteq S$, thus, $q\in S^{0}$. Since $q$ was an arbitrary point in $(p-c,p+c)$ it follows that $(p-c,p+c)\subseteq S^{0}$ and so $p$ is an isolated point of $S^{0}$. Therefore, $S^{0}$ is open.

$\Box$

Reflection: Straight from the definitions.