Suppose $X,\tau$ and $Y,\tau '$ are contractible to $x_0$ and $y_0$ respectively. Prove that $X\times Y$ is contractible and simply connected. Here is what I got so far.
Assume that $X,\tau$ and $Y,\tau '$ are contractible to $x_0$ and $y_0$ respectively.
Since $X,\tau$ is contractible to $x_0$ 
$$id_X \sim f(x)=x_0 $$
meaning there exist a map $H:X\times [0,1] \to X$ such that
$$H(x,0)=id_X$$
$$H(x,1)=x_0$$
similarly, $Y,\tau'$ is contractible to $y_0$ 
$$id_Y \sim g(x)=y_0 $$
meaning there exist a map $H':Y\times [0,1] \to Y$ such that
$$H'(y,0)=id_Y$$
$$H'(y,1)=y_0$$
Since $X\times Y =\{(x,y): x\in X, y\in Y\}$ so $id_{X\times Y} = X\times Y = image(id_x) \times image(id_Y)$ and now I don't know where I'm going with these useless fact
 A: With your $H$ and $H'$ define $H'': (X \times Y) \times [0,1] \rightarrow X \times Y$ by $H''((x,y), t) = (H(x,t), H'(y,t))$. 
A map into the product $X \times Y$ is continuous iff its compositions with the two projections (onto $X$ and $Y$) are both continuous. And these compositions are just $H$ and $H'$ essentially, so continuous by assumption. This shows that $H''$ is continuous.
And $H''(x,y), 0) = (H(x,0), H'(y,0)) = (x,y)$ for all $(x,y) \in X \times Y$, and also $H''((x,y), 1) = (H(x,1), H'(y,1)) = (x_0, y_0)$ as well. So this defines a contraction of $X \times Y$ to the point $(x_0, y_0)$.
And contractible spaces are simply connected. If you're not allowed to use that, you can use a modification of the above argument to show that a closed loop (w.r.t. a basepoint $(p, q)$) in $X \times Y$ is also homotopic to the constant map to that base point. 
A: Alternative proof requiring familiarity with categories.
The categories $\mathbf{Top}$ and $\mathbf{hTop}$ both have topological
spaces as objects and a topological space is contractible as object
of $\mathbf{Top}$ if and only if it is terminal as object of $\mathbf{hTop}$.
Secondly quotientfunctor $\left[\right]:\mathbf{Top}\rightarrow\mathbf{hTop}$
preserves products. 
Thirdly the product of terminal objects is a terminal object.
Now the (short and elegant) proof:
$X,Y$ contractible in $\mathbf{Top}$ $\Rightarrow$ $X,Y$ terminal
in $\mathbf{hTop}$ $\Rightarrow$ $X\times Y$ terminal in $\mathbf{hTop}$
$\Rightarrow$ $X\times Y$ contractible in $\mathbf{Top}$.
It is not unthinkable that this is not (yet) of much use to you. However, see it as an encouragement to explore categories. It can be very enlightening.
