Suppose $A_{1}$ and $A_{2}$ are topological spaces. Define the product topology on $A_{1} \times A_{2}$. Let $\pi_{i}: A_{1} \times A_{2} \rightarrow A_{i}$ be the projection. Show that a map $F: X \rightarrow A_{1} \times A_{2}$ is continuous if and only if $\pi_{1} \circ F$ and $\pi_{2} \circ F$ are continuous.

Prove that if $A_{1}$ and $A_{2}$ are connected, then $A_{1} \times A_{2}$ is connected.

Let $X$ be the topological space whose underlying set is $\mathbb{R}$, and whose open sets are of the form $(a, \infty)$ for $a \in \mathbb{R}$, along with the empty set and the whole space. Describe the open sets in $X \times X$. Are the maps $f, g: X \times X \rightarrow X$ defined by $f(x, y)=x+y$ and $g(x, y)=x y$ continuous? Justify your answers.