I have Cantor intersection theorem:
Let $X$ be a complete metric space, and let $\{F_n\}$ be a decreasing sequence of non-empty closed subsets of X such that $d(F_n)$ converges to $0$.
Then $F=\bigcap_{n=1}^\infty F_n$ contains exactly one point.
What affects whether the condition $d(F_n)$ converges to zero is removed?
Give an example to show that the set $F$ in the intersection of Cantor theorem may be empty if this happens?
[Math] Cantor intersection theorem
general-topology
Best Answer
Note: The following examples show that the conditions $\displaystyle \lim_{x\to\infty} d(F_{n})=0$ and that $F_{n}$ are closed sets both are necessary for the validity of the theorem.
Example: Let $X$ be the real line $R$ and let $F_{n}=[n,\infty).$ Now we know that $X$ is complete, $F_{1}\supset F_{2}\supset F_{3}....$ and $F_{n}$ are closed sets. But $\displaystyle \bigcap_{n=1}^{\infty} F_{n}=\phi.$Note that $\displaystyle \lim_{n\to\infty}d(F_{n})\neq 0.$
Example: Let $X$ be the real line $R$ and let $\displaystyle F_{n}= ( 0, \frac{1}{n}].$ Now we know that $X$ is complete, $F_{1}\supset F_{2}\supset F_{3}....$ and $\displaystyle \lim_{n\to\infty}d(F_{n})=0.$ But $\displaystyle \bigcap_{n=1}^{\infty} F_{n}=\phi.$ Note that the $F_{n}'s$ are not closed.