Let $(M,d)$ be a metric space and $B(x, \epsilon) = \{ y \in M \mid d(x,y) < \epsilon \}$ and $\bar{B}(x, \epsilon) = \{ y \in M \mid d(x,y) \leqslant \epsilon \}$. In general, it is not true that $\bar{B}(x, \epsilon) = \overline{B(x, \epsilon)}$ (closure of $B(x, \epsilon)$).
The metric space $M$ is said to have:
- nice closed balls if and only if $\bar{B}(x, \epsilon) = M$ or $\bar{B}(x, \epsilon)$ is compact for every $x\in M$ and $\epsilon \in \mathbb{R}^+$.
Compact closed balls if and only if the ball $\bar{B}(x, \epsilon)$ is compact for every $x\in M$ and $\epsilon \in \mathbb{R}^+$.
Let $N$ be a metrizable topological manifold (no additional structure). Questions:
It is true in $N$ that $\bar{B}(x, \epsilon) = \overline{B(x, \epsilon)}$ ? Does $N$ has nice closed balls and compact closed balls ?
Thanks in advance !! Cheers...