Is there a general rule of what kind of sets it is easier to prove connectedness using path connectedness or regular connectedness?
I understand that path connected $\implies$ connected, but are there situations where it's easier to prove using path connectedness vs. connectedness and vice-versa? One example I know of is that proving that an interval is connected in $\mathbb{R}$ by constructing a function
$\gamma:[0,1]\rightarrow [a,b]: \gamma(t)=c+t(d-c), t\in [0,1]$
which is clearly continuous. The connectedness approach is a lot more laborious. So my question is, is there a general rule of what kind of sets it is easier to prove connectedness using path connectedness or regular connectedness?