[Topological Basics for Optimizations](Topological%20Basics%20for%20Optimizations.md) --- ### **Interior** The interior of a set $B\subseteq \mathbb R^n$ is the set of all the point such that for each point, there exists an closed epsilon ball surrounding that point and the whole ball is in the set $B$. Denotes the interior of the set as $\text{int}(B)$, then we have the definition that: #### **Def | Interior of a Set** > $ > x\in \text{int}(B)\iff > \exists \delta > 0: \mathbb{B}_\delta \subset Q > $ **Context** The characterization allows moving around the point a tiny bit, make things easier when dealing with infinity. **Observations** If $C_1 \subseteq C_2$, then it's interior still preserves this relation **Facts** We call a set $C$ is open if $\text{int}(C) = C$, for notation simplicity we make $\text{int}C = C^\circ$. #### **Definition | Relatively open** > Let $(M,d)$ be a metric space. > Let $X \subseteq M$. > The set $O_X$ is relatively open in $X$, if there exists open set $O \subseteq M$ such that $O \cap X = O_X$. **Examples** Suppose $M = \R$, and let $X = [0, 1]$. Then the set $O_X = [0, 1)$ is relatively open in $X$. Consider any $\epsilon > 0$ and $O = (-\epsilon, 1)$ then $O \cap X = (-\epsilon, 1)\cap [0, 1] = [0, 1) = O_X$. Therefore, the set $[0, 1)$ is relatively open in $[0, 1]$. #### **Thm | Maximal Open set Subset is the Interior of the Set** > The interior of the set $C$ is the union of all open set that is contained in $C$, meaning that $\text{int}(C) = \bigcup\{S\subseteq C | S^\circ = S\}$ **Proof** If $x \in C^\circ$ then exists open ball $\mathbb = \{x : \Vert x\Vert < 1\}$, $\epsilon > 0$ where $x + \epsilon \mathbb B \subseteq C$, since $x + \epsilon \mathbb B$ is an open set, it belongs to the maximal open subsets of $C$. Choose any $x \in \bigcup\{S\subseteq C | S^\circ = S\}$, then by some type of axiom (maybe the axiom of choice) we have $x \in S \subseteq S$, and by the axiom of topology, there exists $\epsilon > 0$ such that $s + \epsilon \subseteq S \subseteq C$, and hence by the above definition of interior we have $x \in C^\circ$.