Connected $G_\delta$ sets in a connected completely metrizable space with more than one point.

Question

Suppose $(X,\tau)$ is a connected completely metrizable space with more than one point. Let $\mathbb{G}$ be the set of all connected $G_\delta$ subsets of $X$. And let $\mathbb{O}$ be the class of connected open subsets of $X$.

(1) Is it possible that $\mathbb{G}$ is finite? (2) Can $\mathbb{O}$ be finite?

Answer 1

If $X$ has at least two points then $X$ is uncountable and since every singleton $\{x\}$ is an $G_{\delta }$ set and connected then $\mathbb{G}$ must be infinite.