Fun question (Metric Spaces)

[marcmtlca]

Find a complete metric space (X,d) and a nested sequence of closed balls in X whose intersection is empty.

\(\displaystyle B_1 \supseteq B_2 \supseteq \cdots \supseteq B_n \supseteq \cdots\)

and

\(\displaystyle \bigcap_{n=1}^{\infty}B_n=\emptyset\)

give it a try (provided you don't already know of the answer immediately)

 

[JakeD]

Find a complete metric space (X,d) and a nested sequence of closed balls in X whose intersection is empty.

\(\displaystyle B_1 \supseteq B_2 \supseteq \cdots \supseteq B_n \supseteq \cdots\)

and

\(\displaystyle \bigcap_{n=1}^{\infty}B_n=\emptyset\)

give it a try (provided you don't already know of the answer immediately)

The set of irrational numbers with the Baire metric is a complete metric space. Take the centers of the closed balls to be a sequence of irrational numbers converging to zero.

The Baire metric is used because the set of irrational numbers with the Euclidean metric is not complete. Do you have an example using a more well-known metric?

 

[marcmtlca]

The answer I came up with was the the set of natural numbers with the metric:

\(\displaystyle d(a,b)\, =\, \frac{1}{a}\, +\, \frac{1}{b}\, +\, 1\)

...when \(\displaystyle a\, \not =\, b\), and:

\(\displaystyle d(a,b)\, =\, 0\)

...when \(\displaystyle a\, =\, b\). You then take the closed balls:

\(\displaystyle B_n\, =\, B(n,\, \frac{1}{n}\, +\, \frac{1}{n\, +\, 1}\, +\, 1)\, =\, \{z \in \bf{N}\, :\, d(z,n)\, \le\, \frac{1}{n}\, +\, \frac{1}{n\, +\, 1}\, +\, 1 \}\)

I tried to read up on the Baire metric but couldn't find anything concrete. Could you write down the metric?

 

[stapel]

Recreational puzzles can be fun, but it might be considerate to keep posts "on topic" in the tutoring forum. Games and riddles should go in "Odds and Ends", so I will move this there.

Thank you for your consideration.

Eliz.

 

[JakeD]

The answer I came up with was the the set of natural numbers with the metric:
\(\displaystyle d(a,b)=\frac{1}{a}+\frac{1}{b}+1\) when \(\displaystyle a \not = b\) and \(\displaystyle d(a,b)=0\) when \(\displaystyle a=b\). You then take the closed balls: \(\displaystyle B_n=B(n,\frac{1}{n}+\frac{1}{n+1}+1)=\{z \in \bf{N} : d(z,n) \le \frac{1}{n}+\frac{1}{n+1}+1 \}\)

I tried to read up on the Baire metric but couldn't find anything concrete... could you write down the metric?

Yours is a nice example. That is an interesting metric.

The Baire metric on the irrational numbers in [0,1] is this: the distance between two irrationals is 1/n when the first n digits of their continued fraction forms match. My example does not work because a nested sequence of closed balls will have a non-empty intersection.

 

[marcmtlca]

Yeah, the tricky part is avoiding the centers.