Let $d\in\mathbb N$ and $B$ be a closed subset$^1$ of $\mathbb R^d$.
Can we always write $B$ as the (finite / countable / uncountable) union of connected (or even path connected) subsets of $B$?
I guess the elements of such a union are what's being called the "connected components" of $B$, but I'm uncertain whether they always exist, if they are finite, countable or countable and which kind of connectedness notion they satisfy when the topological space in question is $\mathbb R^d$ (or maybe, more generally, a metric space).
$^1$ .I'm actually interested in the case where $B$ is the boundary of another closed subset of $\mathbb R^d$, but I guess this won't matter in the following. However, if I'm wrong, feel free to assume that.
Any subset of $\mathbb R^{d}$ is a union of singleton sets which are connected.