Strongly decreasing sequence of sets

**hjortur** · November 14th 2010, 03:50 PM

This was a homework question few weeks ago, and the teacher didn't give
out the solution to this problem (it doesn't directly concern complex analysis), so I was wondering weather someone knows the answer to this one.

Let $A\subset \mathbb{C}$ , and $D$ be such that
$D(A)$ is the set $A$ with all isolated points of $A$ removed. So the question is, is there a set $A\subset \mathbb{C}$
such that the sequence, $(D^{n}(A))_{n\in\mathbb{N}}$ is strongly decreasing?
That is $D(A)\subset A$ , $D^{2}(A)\subset D(A)$ , but not equal?

My first thoughts went something like this: Let $M:=\{\frac{1}{n}:n\in\mathbb{N}\}\cup\{0\}$ . Then $D(M)=\{0\}$ and
$D^{k}(M)=\emptyset$ if $k\geq 2$ , so this wont do.

Now for every $x\in M$ $x\ne0$ we can make $\varepsilon$ -balls such that $\varepsilon_{x_1}\cap\varepsilon_{x_2}=\emptyset$ if $x_1\ne x_2$ and neither is 0. Then we can construct a sequence
for every $x\in M$ such that its limit is $x$ and it is entirely contained in $\varepsilon_x$ . Lets call the set of those sequences $S_M$ .
Now let $M_1=M\cup S_M$ .
$M_1$ is such that $D(M_1)=M$ so now we got 4 strictly decreasing items in the sequence.

For every given $n\in\mathbb{N}$ we can continue this construction
such that the sequence is strictly decreasing for the first $n$ terms.
But that is not the same as the infinite sequence is strictly decreasing.

I am starting to think that this is impossible, but I am not sure.
Does anyone have any ideas?

**Drexel28** · November 14th 2010, 04:50 PM

Originally Posted by hjortur

This was a homework question few weeks ago, and the teacher didn't give
out the solution to this problem (it doesn't directly concern complex analysis), so I was wondering weather someone knows the answer to this one.

Let $A\subset \mathbb{C}$ , and $D$ be such that
$D(A)$ is the set $A$ with all isolated points of $A$ removed. So the question is, is there a set $A\subset \mathbb{C}$
such that the sequence, $(D^{n}(A))_{n\in\mathbb{N}}$ is strongly decreasing?
That is $D(A)\subset A$ , $D^{2}(A)\subset D(A)$ , but not equal?

My first thoughts went something like this: Let $M:=\{\frac{1}{n}:n\in\mathbb{N}\}\cup\{0\}$ . Then $D(M)=\{0\}$ and
$D^{k}(M)=\emptyset$ if $k\geq 2$ , so this wont do.

Now for every $x\in M$ $x\ne0$ we can make $\varepsilon$ -balls such that $\varepsilon_{x_1}\cap\varepsilon_{x_2}=\emptyset$ if $x_1\ne x_2$ and neither is 0. Then we can construct a sequence
for every $x\in M$ such that its limit is $x$ and it is entirely contained in $\varepsilon_x$ . Lets call the set of those sequences $S_M$ .
Now let $M_1=M\cup S_M$ .
$M_1$ is such that $D(M_1)=M$ so now we got 4 strictly decreasing items in the sequence.

For every given $n\in\mathbb{N}$ we can continue this construction
such that the sequence is strictly decreasing for the first $n$ terms.
But that is not the same as the infinite sequence is strictly decreasing.

I am starting to think that this is impossible, but I am not sure.
Does anyone have any ideas?

So, you're asking does there exist a set $A\subseteq\mathbb{C}$ so that if $D(A)$ is the set of limit points of $A$ then

$A\supset D(A)\supset D\left(D\left(A\right)\right)\supset D\left(\left(\left(A\right)\right)\right)\supset\c dots$

What makes sense to me is to go a route similar to what you did. So, let's think of, just for now, a subset of the real numbers which survives two applications of $D$ . Something like $D=[-5,-4]\cup\{0\}\cup\left\{\frac{1}{n}:n\in\mathbb{N}\rig ht\}$ . What about a set which survives three? What if we took now the subset of the complex numbers which has the real line, appended to the vertical line $\left\{\frac{i}{n}:n\in\mathbb{N}\}$ and then for each $n_0\in\mathbb{N}$ we considered $\frac{i}{n_0}+\frac{1}{m}$ for all $m\in\mathbb{N}$ . Then, I feel that $D\left(A\right)=\mathbb{R}\cup\left\{\frac{i}{n}:n \in\mathbb{N}\right\}$ and $D\left(D\left(A\right)\right)=\mathbb{R}$ . What if you added to each point of the $\frac{i}{n_0}+\frac{1}{m}$ a sequence converging to it, and to each one of those points append a, etc.

I don't feel this is correct though. Let me think about it.

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