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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!

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    What do yo mean by the bar over $\mathbb R$?2012-12-26

2 Answers 2

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As you probably know, the usual topology on $\overline{\Bbb R}$ is the one that makes it homeomorphic to $[0,1]$ in the obvious way: basic open nbhds of $+\infty$ are sets of the form $(x,+\infty]$, and basic open nbhds of $-\infty$ are sets of the form $[-\infty,x)$, for $x\in\Bbb R$. The only reasonable topology to put on $\overline{\Bbb R}^2$ is the usual product topology. I’ve not seen it explicitly used very often, but it is used in this paper, for instance, to define some generalized gauge (Kurzweil-Henstock) integrals of functions of two variables.

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Remember that each point $\bar{\mathbb{R}}\times\bar{\mathbb{R}}$ is an ordered pair $(x,y)$. We define the two projection maps on the $x$ and $y$ axis respectively as $\pi_x,\pi_y:\bar{\mathbb{R}}\times\bar{\mathbb{R}}\to \bar{\mathbb{R}}$ and $\pi_x (x,y)=x$ and $\pi_y (x,y)=y$ The (Tychonoff) product topology for $\bar{\mathbb{R}}\times\bar{\mathbb{R}}$ is the smallest topology on $\bar{\mathbb{R}}\times\bar{\mathbb{R}}$ such that both $\pi_x,\pi_y$ are continuous.

Continuity of the projections obviously depends on the topology of $\bar{\mathbb{R}}$. If you take the usual topology of $\bar{\mathbb{R}}$ then you end up with the usual topology of $\bar{\mathbb{R}}\times \bar{\mathbb{R}}$