Proof Verification: Finding A Ball Strictly Contained In An Open Set Of A Metric Space Problem: Let $X$ be a metric space and let $A$ be an open set of $X$ containing a point $x \in X$. Prove that there exists an $\epsilon > 0$ such that $B_{\epsilon}(x)$ is strictly contained in $A$.
Proof Attempt:
Case 1: $\partial A = \emptyset$
Since $X$ is a metric space, this implies that $A$ is clopen. The only clopen sets of a metric space are $\emptyset$ and the entire space. $A$ contains $x$, so it cannot be empty and thus $A = X$, so any $\epsilon > 0$ will suffice.
Case 2: $\partial A \neq \emptyset$
Let $\displaystyle \epsilon = \frac{1}{2}\inf_{p \in \partial A}{d(p,x)}$, where $d$ is the metric of $X$. Note that $\epsilon \neq 0$ or else this would imply that $x \in \partial A$, which contradicts the hypothesis that $A$ contains $x$ and that $A$ is an open set. So $\epsilon > 0$. Then $B_{\epsilon}(x)$ is strictly contained in $A$ (I'm not sure how to justify this part). $\blacksquare$
Is this proof correct? How do I finish the proof? Thanks.

What's your definition of open?
Mine is a set, $A$, open if every point is an interior point.  And my definition of interior point, $x$,  is if there is an open ball with $x$ at its center that is completely contained in $A$.
Which makes this statement true by definition.
By your given proof it seems you are using a definition: $A$ is open if it has no boundary points and a boundary point, I am assuming, is defined as a point $x \in X$ so that every open ball of $x$ contains points that are in $A$ and points that are not in $A$.
Okay... so assume $A$ is open and $x\in A$.  Then $x$ is not a boundary point because no boundary points of $A$ exist. Now every open ball centered on $x$ contain $x \in A$ so every open ball centered on $x$ contains a point in $A$.  And as $x$ is not a boundary point:


*

*It is not true that every open ball centered at $x$ will contain points in $A$ and will contain points not in $A$.

*Therefore there will exist an open ball, $B_\epsilon(x)$, for which it is not true that will contain both points in $A$ and points not in $A$.

*Therefore for $B_\epsilon(x)$ it will be true that either all points are in $A$ or that all points are not in $A$.

*However $x \in A$ and $x \in B_\epsilon(x)$ so it is not true that all points are not in $A$.

*So it is true that all points in $B_\epsilon(x)$ are in $A$.


So $B_\epsilon(x)$ is completely contained in $A$.
