For given ponit $x$ in a metric space there is some open ball about $x$ such that closure of smaller ball about $x$ contained in that ball.

Question

Let $(M,d) $ be metric space and $x$ is in $M$. Can we find some open ball $B(x,r)$ such that closure of each balls $ B(x,s)$ contained in $B(x,r)$ whenever $ s <r$?

Answer 1

Well, you may proceed this way.

Let $B \left( x, r \right)$ be given. Let $s < r$. Then, we need to show that $\overline{B\left( x, s \right)} \subseteq B \left( x, r \right)$. So, let $y \in \overline{B \left( x, s \right)}$.

Then, $\forall \epsilon > 0, \exists z \in B \left( x, s \right)$ such that $d \left( y, z \right) < \epsilon$. Hence,

$$d \left( x, y \right) \leq d \left( x, z \right) + d \left( y, z \right) < s + \epsilon$$

Hence, $d \left( x, y \right) \leq s < r$ and $y \in B \left( x, r \right)$. Hence the proof!