Product Spaces & Homeomorphism

I have a question that I can't start. I think if I got the start of it I could finish it but I dont know what my first move would be

Let $\displaystyle X,T$ be the product space $\displaystyle \pi(X_i,T_i)$. Given any 'origin' $\displaystyle \bar{x}=(\bar{x_i})_{i+1}$ in X and any $\displaystyle i_0 \in I$.

Show that the map

$\displaystyle \mu: x_i_{0} \rightarrow X$ given by

$\displaystyle \mu(y) = (x_i)_{i+1}$ where $\displaystyle x_i = \bar{x_i}$ if $\displaystyle i$ is not equal to $\displaystyle i_0$ or $\displaystyle x_i = y$if $\displaystyle i=i_0$

is a homeomorphism between $\displaystyle x_i_{0}$ and $\displaystyle \mu(x_i_{0})$