Let $X$ be a complete separable metric space containing no perfect set of size greater than $1$. In other words every subset of $X$ has an isolated point.
It is well known that $X$ must be countable. It follows that $X$ is zero dimensional.
I also discovered that $X$ has a dense discrete subset: Let $\{U_n:n\in\omega\}$ be a basis for $X$. For each $n\in\omega$ there exists $p_n\in U_n$ such that $p_n$ is isolated in $U_n$, and therefore $p_n$ is isolated in $X$. Then $\{p_n:n\in\omega\}$ is a dense discrete subset of $X$.
By similar arguments, $X$ is scattered, meaning for every closed $C\subseteq X$ the set of isolated points of $C$ is dense in $C$. This argument does not use completeness at all.
Can you think of any other interesting properties that $X$ must have?
Can you give any interesting examples of such spaces? The only ones I can think of are discrete spaces and countable unions of convergent sequences.
