Metric and Topological Spaces

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## Exercises 2

1. Prove that the metrics d1 , d2 and d on R2 do indeed satisfy the triangle inequality. What about the corresponding metrics on Rn ?

2. Define a metric on the unit circle in R2 by d(X, Y) = the length of the arc XY. Hence or otherwise prove that the formula d(x, y) = min{ |x - y|, 1 - |x - y| } defines a metric on the open unit interval (0, 1). Does the same formula define a metric on the closed interval [0, 1] ?

3. Let p be a prime number. Define a function f on the positive integers N by the formula
f (n) = pk where this is the largest power of p dividing n and a function g on the rationals Q by g(a/b) = f (|a|) / f (|b|) so that g(r) is the "largest power of p dividing r".
Prove that the function d given by d(r, s) = 1/g(|r - s|) defines a metric on Q.
[This is called the p-adic metric.]
Prove that in this metric the sequence (p, p2 , p3 , ... ) converges to 0.

4. If P and Q are any two points in R2, prove that d1(P, Q) d2(P, Q) d(P, Q).
Hence prove that if a sequence (a1 , a2 , ...) in R2 is convergent in the metric d1 then it is also convergent in d2 and d.
Prove also that d1(P, Q) 2 d2(P, Q) and d2(P, Q) 2 d(P, Q).
Deduce that if a sequence (a1 , a2 , ...) in R2 is convergent in one of the metrics d1 , d2 , d then it is convergent in all of them.

5. Which of the following real-valued functions on the open interval (0, 1) are continuous?

1. Define a function f on a real-number x by taking the decimal expansion of x (terminating in infinitely many 0's rather than infinitely many 9's if it is an exact decimal) and discarding the first, third, fifth and so on, decimal places.
So, for example, f( 0.1234) = 0.24, f (0.1415926536...) = 0.45256... .

2. Define a function g on a real-number x by taking the decimal expansion of x and replacing 0's by 1's, replacing 1's by 2's and so on except that 9's are replaced by 0's.
So, for example, g(0.1298) = 0.23091111... (since the infinitely many 0's at the end all get replaced), g(0.1415926536...) = 0.2526037647... .

3. Define a function h on a real-number x by taking the decimal expansion of x and replacing 0's by 9's, replacing 1's by 8's , 2's by 7's, 3's by 6's, 4's by 5's and vice-versa.
So, for example, h(0.1298) = 0.9701999999... = 0.9702 (since the infinitely many 0's at the end all get replaced), h(0.1415926536...) = 0.9594073463... .

[Hint: Observe that the sequence (0.49, 0.499, 0.4999, ...) converges to 0.5 and use the fact that continuous functions map convergent sequences to convergent sequences.]

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JOC February 2004