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?
Best Answer
In my experiences, in metric spaces, path connectedness is usually-not always,but usually- easier and gives connectedness easily. In general topological spaces,it's sometimes not so easy to construct paths and the general definition of connectedness is usually more useful.
Interestingly, there are cases of topological spaces where it is fairly easy to prove the space is connected,but difficult to prove it is path connected. For example,the topologist's sine curve is connected but not path connected. How do we derive path connectedness given connectedness? The following theorem answers the question:
Theorem: Let S be a subset of a topological space X. Then if S is connected and locally path connected, then S is path connected.
The proof is somewhat lengthy,but not difficult. (Hint to get you started: Consider a path component S of X. By the definition of local connectedness, S is open in X. But what about the complement of S in X? Since X is locally path connected, there is at least one open set U for every point in the complement of S in X which is connected. So now consider the union of U and S in X.)
Notice the converse is NOT true, although it's easy to erroneously assume it is. The comb space is path connected but NOT locally path connected: http://en.wikipedia.org/wiki/Comb_space