# connectedness vs. path connectedness

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?

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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 prove $[0,1]$ is connected anyhow to say that path-connected implies connected. Perhaps the most important class of spaces for which connected implies path-connected is manifolds. – Brett Frankel Mar 9 '12 at 4:01
I can use the fact that path connected implies connected without proof in my class. So I was wondering of what were some examples where this might come in handy -- where I am asked to prove some set is connected and use path connectedness instead – Emir Mar 9 '12 at 5:05
@jay: No, Emir is asking whether there is asking whether there is a good heuristic for deciding which of the two is easier to prove for spaces that are path-connected. – Brian M. Scott Mar 9 '12 at 9:05