Collection of continuum subsets

[jostpuur]

Let's define a set (collection) $\mathcal{C}$ by the following conditions.

$X\in\mathcal{C}$ iff all following conditions hold:

1: $X\subset [0,1]$.

2: $X$ is closed.

3: If $x\in X$ and $x<1$, then there exists $x'\in X$ such that $x<x'$.

4: For all $x\in X$ there exists a $\delta_x >0$ such that $]x,x+\delta_x[\;\cap\; X=\emptyset$. (So $x$ is not "cluster point from right".)

For example, if

$$X=\big\{1-\frac{1}{n}\;\big|\; n\in\{1,2,3,\ldots\}\big\}\;\cup\;\{1\}$$

then $X\in\mathcal{C}$.

It is easy to construct all kinds of members of $\mathcal{C}$, but they all seem to be countable. My question is that does there exist an uncountable member in $\mathcal{C}$?

 

[economicsnerd]

No; every element of $\mathcal C$ is countable. Consider any $X\in\mathcal C$, and fix a function $\delta:X\to(0,\infty)$ as guaranteed by (4). For any $n \in \mathbb N,$ let
$$X_n:= \left\{x\in X: \enspace \delta_x \geq \frac 1 n\right\}.$$
As $(x, x+\delta_x)\cap(y,y+\delta_y)=\emptyset$ for any distinct $x,y\in X_n,$ it follows (from a pigeonhole argument) that $$1\geq \sum_{x\in X_n}\delta_x\geq\sum_{x\in X_n}\frac 1 n = \frac 1 n |X_n|,$$ i.e. $|X_n|\leq n.$ In particular, $X_n$ is finite. Then, as a countable union of finite sets, $$X = \bigcup_{n=1}^\infty X_n$$ is countable.

 

[jostpuur]

Nice proof. I was hoping that an uncoutable member would have existed, so that's why I didn't succeed in proving this myself...

 

[jostpuur]

Yes it is all clear. A collection of disjoint intervals is always countable. Now $X$ has the same cardinality as a collection of disjoint intervals. I must have been distracted by the other details.

 

[jostpuur]

I was hoping that I could define a mapping that maps all countable ordinal numbers into some set of $\mathcal{C}$. So the result was that this cannot be done. How unfortunate. But if you look at the first countable ordinal numbers, it would seem an intuitively reasonable feat.

 

[economicsnerd]

Yeah, it's weird. It seems there exists an element $X_\alpha \in \mathcal C$ which is order-isomphoric to $[0,\alpha]$ for any countable ordinal $\alpha,$ but not for any uncountable one (even the first uncountable one).