Let X be a set, and d1, d2 be metrics on X. Then it is true...

Let X be a set, and d_1, d_2 be metrics on X. Then it is true that d_3 = d_1 + d_2 is also a metric on X (you may assume this to be the case). To be precise, for x, y ? X we define d_3(x, y) = d_2(x, y) + d_1(x, y). Find - with proof - an example of X, d_1, d_2 such that the metric topology induced by d_3 is distinct from the metric topology induced by either d_1 or d_2.

Transcript

00:01 Hello, let x be the set of integers.

00:05 Now embed x in r into two ways.

00:09 In the first way, let's call this embedding phi1.

00:12 The even, the odd integers goes to the odd integers, but the even integers map to 1 over 2n when that even integer is different from 0 and it maps to 0 when 0 maps to 0.

00:25 So, and define another embedding of x integers into r by mapping 2n plus 1 to 1 over 2n plus 1, mapping the odd numbers to 1 over the odd integers to 1 over the odd integer and the even number map to themselves.

00:46 Now define the metric d1 as d1 between two integers mn, the distance under d1 between two integers mn is equal to distance between phi1m and phi1n in r.

01:05 So, this is the induced metric on x induced through phi1.

01:10 Similarly, define the metric d2 on x by setting d2 mn equal to the distance between the phi2m and phi2n in r.

01:32 And yeah, so let us see some properties of the d1 metric.

01:40 Now, if, yeah, now see that the distance between 0 and 2n in the d1 metric is 1 over 2n or the distance between and the distance between 0 and an odd integer is the magnitude of the odd integer.

02:08 But the distance between 0 and odd integer is equal to 1 over the magnitude of the odd integer in the d2 metric and the distance of between and the distance between 0 and even integer in the d2 metric is the magnitude of the even integer.

02:30 So, in the d2 metric, this sequence of numbers 1, 3, 5, 7, they converge to 0 while this sequence of numbers 0, 2, 4, 6 is not cauchy while in the d1 metric this sequence of numbers 2, 4, 6 under d1 metric they converge to 0 but this sequence of numbers 1, 3, 5, 7 is not cauchy.

02:57 Now, let us define d3 on x, third metric on x, d3 mn is equal to d1 mn plus d2 mn.

03:11 Okay, now consider let xn be the sequence 1, 3, 5 goes on the odd positive odd numbers.

03:20 Now, xn converges to 0 in the d2 metric but xn is not cauchy in particular does not converge to 0 in the d1 metric.

03:29 Similarly, if you consider the sequence of non -negative even integers 0, 2, 4, 6 and let's call it yn, yn converges to 0 in the d1 metric but yn is not cauchy in particular it does not converge to 0 in the d2 metric.

03:48 Now, the distance between 0 and xn is equal to in d3 metric is equal to the sum of the distance between 0 and xn in the d1 metric and the distance between 0 and xn in the d2 metric...

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