Has this "thinner" Cantor set been defined and studied before?

Question

The Cantor set is defined by starting with the closed interval $[0,1]$ and repeatedly taking "bites" of open intervals in the middle, like $(1/3,2/3)$. However, I have thought of something like a "thinner" Cantor set. We start with the open interval $(0,1)$ and repeatedly take "bites" of closed intervals in the middle, like $[1/3,2/3]$. It is easy to see that this thin Cantor set is a proper subset of the regular Cantor set. I want to know, has this set and its properties been studied before in the mathematical literature? I mainly want to know whether this set of real numbers is open, closed, or neither.

Answer 1

Let the coboundary of a set $A$ (in some "background" topological space, here $\mathbb{R}$ with the usual topology) be the boundary of the complement of $A$. For example, every point in the Cantor set $C$ is a boundary point of $C$, but $C$ has only countably many coboundary points, and these are exactly the endpoints of the intervals in the usual "stage approximation" of $C$. (Unfortunately, I don't know what this is actually called.)

Your modification $\widehat{C}$ of $C$ is just $C$ minus its coboundary; note that since the complement of $C$ is open and every open set in $\mathbb{R}$ has countable boundary, we get that $C\setminus \widehat{C}$ is countable. In particular, $\widehat{C}$ is nonempty. Since $C$ (so a fortiori $\widehat{C}$) has empty interior, $\widehat{C}$ is not open, and it's clear $\widehat{C}$ is not closed since its closure is just $C$ itself. On the other hand, as a closed set minus a countable set $\widehat{C}$ is $G_\delta$ (it's the intersection of a closed set and a cocountable set; closed and cocountable sets are each $G_\delta$, and the intersection of $G_\delta$ sets is $G_\delta$).

Beyond this, I don't think there's much to say.

Answer 2

To answer your question, you may want to check out why the traditional Cantor set is important in the first place. Here is a post that describes how Cantor sets appear in dynamics. In particular, the usual definition we use in dynamical systems is that in a metric space $X$, a subset $C \subseteq X$ is called a Cantor set if $C$ is compact, perfect (i.e., closed and contains no isolated points), and totally disconnected (i.e., the only connected subspaces of $C$ are the empty set and single points).

Another way you could define a Cantor set is by using the shift spaces from "symbolic dynamics." As it turns out, the traditional Cantor set is homeomorphic to the one-sided shift space on two symbols. So you could say a topological space is Cantor if it is homeomorphic to some shift space.

Your proposed set is not Cantor by either definition. Your set is not closed, therefore not compact (the endpoints of the interval "bites" are in the traditional Cantor set, but not in yours). Your set is also not perfect (although I'm guessing your set has no isolated points). But since your set is a subspace of a Cantor set, we could at least say it is totally disconnected and has Lebesgue measure zero.

As for your set having a specific name or a particular lens of study, I am not sure. I would guess not, only because we already know a great deal about its topological properties from the above discussion, and your set won't arise naturally from any simple dynamical systems.