For example, when $a,b$ are reals and $f$ is a real function on $\mathbb{R}$, If $\lim_{x\to a} f(x,y), \lim_{y\to b} f(x,y)$ exist, then $\lim_{(x,y)\to (a,b)} f(x,y)$ exists.
This makes sense to me, but i don't understand how things like $\int_{-\infty}^{\infty} f d\alpha$ can be well defined.
What is the usual topology on $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$?
Thank you in advance!