# path connectedness

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• April 10th 2009, 11:54 AM
Andreamet
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.
• April 10th 2009, 03:20 PM
JaneBennet
Try the map $p:[0,1]\to X,$ $p(t)=a$ if $t\ne1$ and $p(1)=b.$

Then $p^{-1}(\O)=\O,$ $p^{-1}(\{a\})=[0,\,1),$ $p^{-1}(X)=[0,\,1]$ which are all open in $[0,\,1].$