I wrote a proof that any path connected space is contractible which is completely wrong but i was not able to see what goes wrong in my proof:
Let $X$ be a path connected space. Let $P$ be point in $X$. Write $r$ the retraction $$r:X\rightarrow P;\;\; x\mapsto P$$ For any $x\in X$, take a path $$\gamma_{p,x}:I\rightarrow X$$ that starts in $P$ and ends in $x$. Now consider the homotopy $$H_t:X\rightarrow X;\;\; x\mapsto \gamma_{p,x}(t)$$ Obviously we have $H_0=i\circ r$ where $i:P\rightarrow X$ the inclusion map and $H_1=id_X$. Hence $i\circ r$ is homotopic to $id_X$ which means that $X$ deformation retracts onto $P$ and in particular that $X$ is contractible.
Now that i'm re-reading what i have written i think i see that the problem is that the choice of the path is not made in a continuous way, because there may be different paths starting in $P$ and ending in $x$, is this the problem? and if yes does that mean that if we are able to choose such a path in a canonical way then our space is contractible, for example the Cone on $X$ is contractible because we can choose canonically the path as the line segment between any point of the cone to the apex? thanks for your help!!