Let $X$ be a separable complete metric space.
I wonder if following properties hold in ZF.
- Limit Compact ⇒ Compact
- Does there exist a function$f$ such that $f(E)$ is closed and $f(E)\subset E$, for every infinite set $E$ in $X$.
If 2 doesn't hold, what if $E$ is Dedekind-Infinite?
It seems if 2 holds, 1 holds immediately. (See Constructing a choice function in a complete & separable metric space.)