Path connected space

• Apr 12th 2012, 09:15 PM
monster
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,
• Apr 13th 2012, 08:31 AM
HallsofIvy
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.
• Apr 16th 2012, 02:30 AM
monster
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 pre-image of the open sets are open?