TITLE: Closed circle as a metric space
QUESTION [6 upvotes]: If I take the closed unit circle, $C$, can I create a metric space $(C,d)$, where $d$ is the Euclidean-metric?
The reason for asking is that my text book says that all metric spaces are open, so:
(i) Is my example impermissible as a metric space? or
(ii) For points around the edge of the circle, $a$, the open balls $B(a,\delta)$ aren't your typical open balls, but rather the bit of the balls that intersects with $C$?

REPLY [5 votes]: Every metric space $X$ is both open and closed in itself; note that $\varnothing$ is by definition both closed and open and that $\varnothing = X^{c}$. So $C$ is open in itself but, when considered as a subset of $\mathbb{R}^{2}$, is closed in $\mathbb{R}^{2}$.
I assume by a circle you really mean a circle rather than a disk. Let $a \in C$ and $\delta > 0$. Note that $B(a,\delta) := \{ x \in \mathbb{R}^{2} \mid |x-a| < \delta \}$ is an open ball that looks like an open disk. The set $\{ x \in C \mid |x-a| < \delta \} = B(a,\delta) \cap C$ is an open ball in $C$ but looks not like a ball.