I asked a similar question on the forum earlier, and felt as if I had the issue clarified, but having come upon the problem again in other situations, it is clear that I'm still confused.
Consider the set $X \subset \mathbb{R} = [a,b] \setminus \{c\}$ for some $c$ such that $a < c < b.$ The textbook says $X$ is disconnected and qualifies its statement by putting forth the "separation" $X = [a,c) \sqcup (c,b].$ With respect to the metric subspace $X$, the two intervals $[a,c),(c,b]$ are clopen. However, they are not with respect to $\mathbb{R}.$ Do you have to qualify "connectedness" with a "respect to," as in, $X$ is connected with respect to $\mathbb{R},$ but disconnected with respect to itself. Can you comment on the connectedness of the natural numbers?
Connectedness is a notion about a topological space, not a relative notion. When one speaks of a connected subset $X$ of a topological space $E$, it means the topological space $X$, endowed with the
induced topology, is connected.