Im trying to show that:
for $X,Y$ topological spaces
$X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point
while $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$.
i tried to use that $Id_X \sim x_0$ by the homotopy $F:X\times I \to Y$ (while $I=[0,1]$)
and let $f\in [X,Y]$ so $f\circ F$ will give us a homotopy of $f$ and the constant function $f(x_0)$
but i dont understand how to show it for a fixed point $y_0$ in $Y$. i imagine it involves a path from $y_0$ to $f(x_0)$
Best Answer
You're almost there. If $\gamma : I \to Y$ is a path from $f(x_0)$ to $y$ (which exists because $Y$ is path connected), then the homotopy $H : X \times I \to Y$ defined by $H(x,t) = \gamma(t)$ is a homotopy from the constant function equal to $f(x_0)$ (aka $x \mapsto (f \circ F)(x, 1)$) to the constant function equal to $y_0$.