Metric space extension Is single point extension of a metric space possible? Let $(X,d)$ be a metric space and $\overline{X}=X\cup \{\overline{x}\}$. Is it possible to find a metric $\overline{d}$ for which $(\overline{X},\overline{d})$ is a metric space and $\overline{d}(x)=d(x)$ for every $x\in X$?

Well, you could,
Let $X$ be the integers and $d$ be the discrete metric $d(a, b) = 1$ if $a \ne b$ and $= 0$ if $ a = b$. The discrete metric induces the discrete topology where the open sets are all possible subsets of $X$.
Now add any non-integer to $X$ with the same (discrete) metric.
But why ?
