Contractibility is product-closed

Statement

Let $X_i$, $i\in I$, be an indexed family of topological spaces. Then the product space, endowed with the product topology, is contractible.

We describe the proof for two spaces; the same idea works in general: Let $X$ and $Y$ be contractible spaces. Then the product space $X\times Y$ is contractible.

Proof

Key idea

Suppose $F:X\times I\to X$ and $G:Y\times I\to Y$ are contracting homotopies for $X$ and $Y$. Then the map $F\times G$ defined as:

$(F\times G)(x,y,t)=(F(x,t),G(y,t))$

is a contracting homotopy for $X\times Y$.

Thus $X\times Y$ is contractible.