
path connectedness
This is a pretty trivial question (I think?). I just want to see anyone do it with a bit of detail to determine if I am on the right track. thanks!
Let X={a,b} and assume that X has the topology T={0, {a},X}. Show that X is path connected in this topology.

Try the map $\displaystyle p:[0,1]\to X,$ $\displaystyle p(t)=a$ if $\displaystyle t\ne1$ and $\displaystyle p(1)=b.$
Then $\displaystyle p^{1}(\O)=\O,$ $\displaystyle p^{1}(\{a\})=[0,\,1),$ $\displaystyle p^{1}(X)=[0,\,1]$ which are all open in $\displaystyle [0,\,1].$