I've written the following "proof" that if $X$ is path-connected then $H_0 (X) = 0$. I know that that's not the case, yet I can't find the mistake in my "proof". Can you please point it out to me? Here is my "proof":
$X$ path-connected $\implies $ any 2 singular 0-simplexes (constant maps) $x,y$ in $X$ are the boundary of a 1-simplex (path) $\sigma$
$\iff \partial \sigma = x - y$
$\implies x,y$ differ by a boundary
$\implies x,y$ are in the same homology class $\{ c \} \in H_0(X)$
\implies H_0 (X) = \{ c \} = 0.
Thanks for your help, it's much appreciated.