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Let $A$ be some set of axis-aligned rectangles in the plane, each pair of which has empty intersection. Prove that $A$ is a countable set.

(An axis-aligned rectangle is a set of the form

$$M = {\{\langle x,y \rangle \in \mathbb{R^2} | a \leq x \leq b , c \leq y \leq d}\}$$ for $a,b,c,d$ such that $a < b$ and $ c < d$.)

Attempt:

I tried using the density of the $\mathbb{Q}$ in $(\mathbb{R},\leq)$, but without any success.

Mike Earnest
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Isaac S
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  • I've edited the question to make the terminology more standard -- a "balanced" rectangle should be "axis-aligned," and I think your term "foreign in pairs" can be rephrased as "each pair having nonempty intersection" or "pairwise disjoint." I've also changed the tags slightly, since this isn't a logic question. Feel free to roll back or comment if you disagree with any of the edits. – Carl Schildkraut Jan 05 '21 at 21:45

1 Answers1

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You're on the right track. See if you can show that every axis-aligned rectangle contains a rational point (a point $(x,y)$ with $x,y\in\mathbb Q$). From this, using that the rationals are countable, why does the problem follow?

Carl Schildkraut
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