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.

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