# About The Cantor Intersection Theorem ("Mathematical Analysis 2nd Edition" by Tom M. Apostol)

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.

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

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.

• ryaron, Thank you very much for your answer. Commented Nov 26, 2022 at 21:05