3:27 AM
Last night dream, there's this fractal looking thing:
Where every point in the black space you see here, when zoomed in, will give you the same figure show here, no matter how far you zoom
In addition, the pattern you see here is not really just the blue dots, but that it is a grid with a nonlinear scale and where the horizontal gridlines intersect the vertical ones are given by the blue dots
That is, it actually look something like this, and every point in the black space, when zoomed in looks just like this grid shown here
In other words, if we consider each quadrant separately, each line is actually some set that looks like this: $\text{some countably infinite sequence bounded from below} \times [0,\infty)$
More accurately, the set $X$ has the following iterated construction: Let $s_0$ be a countably infinite sequence with infimum at $0$, then:
\begin{align}
X_0 & = s_0\\
X_n & = X_0 \times [0,\infty)\\
X & = \lim_{n\to \infty} X_n
\end{align}
typo: $X_{n+1} = X_n \times [0, \infty)$
So for example, we can have points at ..., 1/100,1/90,1/80,1/70,1/60,1/50,1/40,1/30,1/20,1/10,1,10,20, .... If we zoom into the interval (1/100,1/90), then we might have something like 1/100 + (..., 1/100,1/90,1/80,1/70,1/60,1/50,1/40,1/30,1/20,1/10,1,10,20, ....), and so on
and it does not matter which interval you choose (including intervals like (a,b) such that $\lim_{a\to b}$), how far you zoom, it will have this exact same sturcture
It is possible this set might be isomorphic to the reals as the above feature will mean effectively the infimum can be any real number, thus making the set complete
Alternately, I think this set can be written as $\{x+\{\frac{1}{y}\},x\in \Bbb{R},y \in \Bbb{N}\}$
