
Path connected space
I'm starting to learn about path connected spaces and am a bit sketchy, any help would be appreciated, cheers,
Lets say i Have the set S = {0,1} with topology { emptyset, {0}, S }
and i need to prove S is path connected and therefore connected,
to show path connectedness,
i need to show a path exists from x to y for any x,y in S,
to show this i need a cotinuous map p : [0,1] > S such that p(0) = x and p(1) = y
im confused how to do this for this set,

Re: Path connected space
Let f(x)= 0 for $\displaystyle 0\le x< 1/2$ and f(x)= 1 for $\displaystyle 1/2 \le x\le 1$.
Show that this function is continuous in this topology.

Re: Path connected space
this is the problem i'm having, do i show this is continuous using epsilon delta? or is that not appropriate here and i should be using defn of continuity where the preimage of the open sets are open?