(Or: "I heard you liked Cantor Sets...")

I'm working on a student project, and the following construction came up very naturally: If $C$ is the usual Cantor Set, build a countable union of copies of this set as follows: start from $C$, and add a one-third scale copy of $C$ in the interval $[1/3,2/3]$, add three one-ninth scale copies of $C$ in the intervals $[1/9, 2/9]$, $[4/9,5/9]$ and $[7/9, 8/9]$, etc.

**To Clarify:** basically, every time you remove an interval in the classical construction, in this version, you do not remove the whole interval but you replace it with an appropriately scaled copy of the Cantor set. But you also add Cantor sets in the "holes of the holes", so to speak. So that's why in the second stage, there is a copy of $C$ in $[4/9,5/9]$, that fills in the middle third gap that appears in the copy of $C$ that was added at the first stage.

I guess the best way to see it is that at step $n$, you add to the set previously constructed copies of $C$ scaled at $3^{-n}$ in **every** empty interval where such a copy will fit.

I have my reasons for wanting to look at this construction, but that got me wondering: it looks so natural that it may very well come up in several contexts.

So. Anyone knows where this construction first appeared? Is it especially notable? Does it illustrate any especially interesting property? I would hate to miss something good about it.

thissmall Cantor set? And then later in the gaps ofthosesmall Cantor sets? – Gerald Edgar Sep 26 '12 at 1:53On inner limiting sets of points in a linear interval, Proceedings of the London Mathematical Society (2) 2 (1904), 316-326. – Dave L Renfro Sep 26 '12 at 18:15