0
$\begingroup$

I am self-studying real and complex analysis by W.Rudin and in a proof it is stated, that every open set $V$ in the plane is a countable union of rectangles $R$.

I have the following definitions at hand and am trying to proof this result.

1) A set $S$ is open iff $S$ is a member of the topology on X.

2) The topology of $\mathbb{R}$ is the set of all unions of segments in $\mathbb{R}$, e.g. sets of the form $(a,b)$.

3) The topology of $\mathbb{R^2}$ is the set of all unions of open circular discs.

4) A rectangle is defined as following $R=I_1\times I_2$, where $I_1,I_2$ are segments in $\mathbb{R}$

I can prove the result by assuming the euclidian metric and the topology resulting from the usual definition of open balls: $\{x \in \mathbb{R}: |a-x|

Help would be very much appreciated.

  • 1
    Open rectangles and open disks generate the same topology, i.e. the "usual" topology on $\mathbb R^2$, that is probably worth verifying. – 2017-01-03
  • 0
    @JustinYoung Thanks for commenting. Can you state a definition of "open disk"? – 2017-01-03
  • 1
    An open disk in the plane is an "open ball" with the usual (euclidean) metric. – 2017-01-03
  • 0
    I don't see how a segment implies the euclidian metric. Can you state a proof, or give a link to a proof? Same thing for the open disk. – 2017-01-03
  • 1
    The "open ball" you defined above is $(a-r, a+r)$. So, open balls are open segments. The other direction you can do from there. – 2017-01-03

1 Answers 1

1

Given any point $z=(x,y)$ of an open set $\Omega\subset{\mathbb R}^2$ there is an $\epsilon$-neighborhood $$U_\epsilon(z):=\{\zeta\in{\mathbb C}\>|\>|\zeta-z|<\epsilon\}\subset\Omega\ ,$$ hence an open rectangle $R_z:=\>]x-h,x+h[\>\times\>]y-h,y+h[\>\subset\Omega$. (Choose, e.g., $h:=\epsilon/2$.) It follows that $\Omega=\bigcup_{z\in\Omega} R_z$.

But we have to do better: It is claimed that we can do with a countable union of rectangles. To this end make each $R_z$ slightly smaller, so that it is of the form $R_z':=\>]a,b[\>\times\>]c,d[\>$ with $a$, $b$, $c$, $d$ rational. There exist only countably many such rectangles, so that $\Omega=\bigcup_{z\in\Omega} R_z'$ is in reality a countable union.

  • 0
    I assume $U_\epsilon (z) = \{w\in \mathbb{R}: d(z,w) < \epsilon \}$, but how can we show, that $R_z$ is indeed in $\Omega$? And what metric is used in the definition of $U_\epsilon (z)$? – 2017-01-03