Take the 2-minute tour ×  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?  share|improve this
question Given that connected doesn't imply path connected, I
guess you're asking for what spaces these properties are
equivalent? –  Lepidopterist Mar 9 '12 at 3:54 Couldn't you just
prove $[a,b]$ is connected by the same proof that $[0,1]$ is
connected? You need to pr