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Let $\{I_k\}$ be a set of nested closed bounded sets such that $\forall k\in\Bbb{N}, I_k\supset I_{k+1}$, and $\lim_{n\to\infty}\operatorname{diam}(I_n)=0$. Let the metric space $(X,d)$ be complete. This implies that $\bigcap_{k=1}^\infty I_k\neq\emptyset$- it contains precisely one point.

Let $J_k=\operatorname{Int}(I_k)$, or the interior of $I_k$. What can we say about $\bigcap_{i=1}^\infty J_k$?

I will list down my assertions one by one:

  1. The infinite intersection of closed sets is a closed set. This is because it is the complement of the infinite union of open sets, which is an open set, and the complement of an open set is a closed set.

  2. So $S=\bigcap_{k=1}^\infty I_k$ is a closed set containing one point. Let that one point be $x$. $x$ can't be the limit point of $\operatorname{Int}(S)$, as $S\setminus \{x\}$ is empty. Hence, $\bigcap_{k=1}^\infty J_k=\bigcap_{k=1}^\infty I_k=\{x\}$, which is a closed point, but not an open point.

Are the arguments sound?

Thanks in advance!

2 Answers2

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The intersection of $J_n$'s may be the point $x$ (when $I_n:=[-n^{-1},n^{—1}]$ on the real line with usual metric) or the emptyset (when $I_n:=[0,n^{-1}]$).

Davide Giraudo
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  • Could you kindly point out where I've gone wrong in my second argument? In your example of $[0,\frac{1}{n}]$, considering intersection of nested open sets, we're talking about $\bigcap_{n=1}^\infty (0,\frac{1}{n})=\emptyset$. We also know that $\bigcap_{n=1}^\infty [0,\frac{1}{n}]=[0]$. Are we saying $\operatorname{Int}[0]=\emptyset$? –  Aug 17 '13 at 12:31
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    Oh god I just realised it is a nowhere dense set. Thanks. My basic confusion was a closed set has to contain limit points. It may contain none, and still be a closed set, as long as it has no external limit points. –  Aug 17 '13 at 12:33
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Since $J_n\subseteq I_n$, we have $\bigcap J_n\subseteq \bigcap I_n$, so $\bigcap J_n$ is either $\{x\}$ or $\emptyset$. Of course the first holds if and only if $x\in J_n$ for all $n$. Note that the intersection of infinitely many open sets need not be open.

Hagen von Eitzen
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