cardinality of a set of non-overlapping rectangles on the plane I was given a question in set theory and wanted to hear you opinions about the answer I had in mind.
The question is this - let T be a set of non-overlapping rectangles on the plane. Find the maximal cardinality of T.
I'm not completely sure about my answer, but I think it's $א$.
My idea was looking at Riemann sums within an integral - it's an infinite set of non-overlapping rectangles, and the width of each rectangle approaches zero, so it seems that there might be א such rectangles inside some closed interval. The problem is that I'm not sure if the cardinality of these rectangles is actually א (and I'm definitely not sure on how to prove this).
Do you think this is a good way to go? Am I even in the right direction? (the cardinality might be smaller and I'm just wasting my time...) How would you tackle this problem?
Thanks in advance.
 A: In part, this depends on the definition of a rectangle.  The usual notion of a rectangle from elementary school geometry is a plane figure with four sides meeting at right angles, so let's first consider that notion.  Also, let's assume that rectangles are not degenerate, i.e. every side has some positive length (e.g. a line segment could be considered a rectangle where two of the sides have length zero—let's exclude that possibility for now).
Let $\mathscr{R}$ be a set of rectangles, such that if $R,S\in\mathscr{R}$ are distinct, then $R\cap S =\emptyset$.  Every such rectangle contains a point with rational coordinates—this follows from the density of $\mathbb{Q}^2$ (the set of points with rational coordinates) in $\mathbb{R}^2$ and the fact that the interior of a rectangle is nonempty (which follows from our assumption that rectangles are non-degenerate).  But then each point with rational coordinates can be contained in at most one rectangle form $\mathscr{R}$.  From this, it follows that
$$ |\mathscr{R}| \le |\mathbb{Q}|, $$
i.e. at best, a collection of disjoint rectangles can be countable.  As it is possible to construct a countable set of disjoint rectangles (consider rectangles of the form
$$ R_n = \left[ n,n+\frac{1}{2} \right]^2, $$
where $n$ ranges over $\mathbb{N}$; the collection $\{R_n\}_{n\in\mathbb{N}}$ is countable, and the elements are mutually disjoint).
On the other hand, we can define rectangles more generally.  For example, we previously dismissed the idea of degenerate rectangles.  If we instead allow degenerate rectangles, then we can find an uncountable set of disjoint rectangles in $\mathbb{R}^2$:  consider the collection
$$ \left\{ \{x\} \times [0,1] : x\in\mathbb{R} \right\}, $$
which is a collection of uncountably many disjoint rectangles.
