I'm trying to solve exercise 6.3#7 from Sidney A. Morris' "Topology without tears": "Prove that each discrete space [...] is a Polish space."
I started by proving that discrete spaces are always completely metrizable with the discrete metric. But then I got stuck. As far as I know, the only dense subset of a discrete space is the whole space.
But does that not mean that only countable discrete spaces are separable (and therefore Polish) spaces?