Is $\{1/n : n = 1, 2, 3, ...\}$ with the subspace topology from $\mathbb{R}$ completely metrizable?
As as result of Baire's category theorem, we know that if a metric space is complete and there are no isolated points, then the space is uncountable. The contrapositive of this says nothing about a complete metric space which is countable and has isolated points (doesn't have to be all isolated points), so we can't make any conclusions about a space of isolated points being completely metrizable or not. It doesn't seem homeomorphic to $\mathbb{R}$, but that also doesn't prove that it's not completely metrizable yet. What else can I use?
This is a countable set, so it cannot possibly be homeomorphic to $\Bbb R$.
However, it is homeomorphic to $\Bbb N$. Simply note that for each $n$, the interval $(\frac1{n+1},\frac1{n-1})$ (or $(\frac23,\frac32)$ for $n=1$) witnesses that each point is isolated. Therefore the subspace topology is discrete, and a countable discrete space is homeomorphic to $\Bbb N$.
In particular it is completely metrizable.