Thread: 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,

Let f(x)= 0 for and f(x)= 1 for .

Show that this function is continuous in this topology.

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 pre-image of the open sets are open?