Does it matter which metric we use on R^2?
Does it change how sequences converge?
Thank you for answering. I was reading something similar. I was mostly confused since their "open sets" are different.
Open sets under the Euclidean metric being circles, and open sets under the taxicab one being squares.
The second bullet point on the page you linked states they are equivalent if the open sets "nest" (are contained in one another)
I can cleary see why this is true, for you can draw a circle inside of a square and a square inside of a circle (as for the values of r', r'' I'm not sure what they'd be)
So they are equivalent
Is there any reason the Euclidean is preferred? Is it just more natural and generalizes easier to higher dimensions? Must every metric be a geodesic?
A metric on induces a topology, so if you can prove that two different metrics induce the same topology on (find a homeomorphism between the two topological spaces), then those metrics are equivalent.