a point metric space X.show {a}^c is open.
Come on you two guys, this is such a fundamental property of metrics.
The distance between two points in a metric space is positive.
$\displaystyle y \in \left\{ x \right\}^c \Rightarrow \quad d(x,y) > 0$
$\displaystyle \left\{ x \right\}^c = \bigcup\limits_{y \in \left\{ x \right\}^c } {B\left( {y;d(x,y)} \right)}$