Question
I need to show the taxicab metric, whihc is given byd(x,y)=(the sum from i=1 to n) of (xi-yi) generates the usualtopology on R^2, given by {U| for every (a,b) in U, there exists apositive delta such that {(x,y) is in R^2| (x-a)^2 +(y-b)^2 is lesthan delta squared} is a subset of U} I'm not even sure how to start this problem, we never learnedhow to generate a topology and our book says this topology isgenerated by the usual metric, not the taxicab. I just need ageneral direction in which to travel. I need to show the taxicab metric, whihc is given byd(x,y)=(the sum from i=1 to n) of (xi-yi) generates the usualtopology on R^2, given by {U| for every (a,b) in U, there exists apositive delta such that {(x,y) is in R^2| (x-a)^2 +(y-b)^2 is lesthan delta squared} is a subset of U} I'm not even sure how to start this problem, we never learnedhow to generate a topology and our book says this topology isgenerated by the usual metric, not the taxicab. I just need ageneral direction in which to travel.
Explanation / Answer
The two topologies are the same if a set is open in one if and onlyif it is open in the other. So, take a set open U in theusual metric. Is it open in the taxicab metric? Take apoint z in U. Is there a > 0 such that all points ataxicab distance less than from z are still in U? Since U is open in the usual sense, there is a disk of positiveradius centred at z which is wholly contained in U. Take = . Using the triangle inequality [sqrt(x2 + y2) sqrt(x2 +y2) >= |x| ] So, in computing the taxicabmetric, each difference in coordinates is less than , so thetaxicab metric is at most 2 = , so any point in ourusual ball of size is in U. So U is open in the usualmetric.