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I am reading "Mathematical Analysis 2nd Edition" by Tom M. Apostol.

Theorem 3.25. Let $\{Q_1,Q_2,\dots\}$ be a countable collection of nonempty sets in $\mathbb{R}^n$ such that:
i) $Q_{k+1}\subseteq Q_{k} \,\,\,\,(k=1,2,3,\dots).$
ii) Each set $Q_k$ is closed and $Q_1$ is bounded.
Then the intersection $\bigcap_{k=1}^\infty Q_k$ is closed and nonempty.

I wondered the following theorem also holds.

Theorem 3.25'. Let $\{Q_1,Q_2,\dots\}$ be a countable collection of nonempty sets in $\mathbb{R}^n$ such that:
i) $Q_{k+1}\subseteq Q_{k} \,\,\,\,(k=1,2,3,\dots).$
ii) Each set $Q_k$ is closed and $Q_1$ is unbounded.
Then the intersection $\bigcap_{k=1}^\infty Q_k$ is closed and nonempty.

But the author didn't write as follows:

Theorem 3.25''. Let $\{Q_1,Q_2,\dots\}$ be a countable collection of nonempty sets in $\mathbb{R}^n$ such that:
i) $Q_{k+1}\subseteq Q_{k} \,\,\,\,(k=1,2,3,\dots).$
ii) Each set $Q_k$ is closed.
Then the intersection $\bigcap_{k=1}^\infty Q_k$ is closed and nonempty.

Please tell me an example such that:

i) Each $Q_k$ is nonempty.
ii) $Q_{k+1}\subseteq Q_{k} \,\,\,\,(k=1,2,3,\dots).$
iii) Each set $Q_k$ is closed.
iv) The intersection $\bigcap_{k=1}^\infty Q_k$ is empty.


My observation:

If some $Q_k$ is bounded, then the intersection $\bigcap_{k=1}^\infty Q_k$ is closed and nonempty.

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    $\begingroup$ Here is the classical example: $$Q_k = [ k ; + \infty )$$ $\endgroup$
    – Crostul
    Commented Nov 26, 2022 at 10:54
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    $\begingroup$ @Crostul Thank you very much for your answer. $\endgroup$
    – tchappy ha
    Commented Nov 26, 2022 at 11:32

1 Answer 1

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First of all, any intersection of closed sets, be it finite, denumerable (countable) or non-denumerable, nested or non-nested is closed.

why?

The union of open sets is open, since let

$x\in\cup_{i\in I}{U_i}$ Then there exists an open neighborhood $B(x)$ of $x$ and an $i\in I$ s.t. $B(x)\subseteq U_i$ and thus $B(x)\subseteq \cup_{i\in I}{U_i}$

Thus, by de-Morgan's law the intersection of closed sets is closed.

Now to address the question of empty or non-empty intersection of nested closed sets:

let $Q_1$ be an unbounded closed and non-empty and let $\{Q_k\}$ be the sequence of closed non-empty sets s.t. $Q_{k+1}\subseteq Q_k$.

The basic idea is to take $R$ and cut finite pieces from it.

Take the space to be $R$ and

$$Q_1=R-(-1,1)\\ Q_2=R-(-2,2)\\ Q_k=R-(-k,k)$$

Each $Q_k$ is closed as it is the complement of an open interval and no $Q_k$ is bounded.

and $(\forall k)(Q_{k+1}\subseteq Q_k)$.

yet the intersection is:

$\cap_{k}{[(x\leq -k)\cup (x\geq k)]}$ so we have two unbounded monotone sequences (one down, one up) that don't have limit points.

So the intersection is empty.

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    $\begingroup$ ryaron, Thank you very much for your answer. $\endgroup$
    – tchappy ha
    Commented Nov 26, 2022 at 21:05

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