December 8th, 2010, 02:42 PM  #1 
Member Joined: Jan 2008 Posts: 34 Thanks: 0  Open Balls...
D(X,Y) = x1y1+x2y2, ?X,Y?R^2 is a metric on the plane of reals How does one show that the family of all open balls in the metric 'D' is a basis for the regular topology on R^2? 
December 8th, 2010, 08:01 PM  #2 
Newbie Joined: Nov 2009 Posts: 3 Thanks: 0  Re: Open Balls...
To show that two bases A and B generate the same topology on X, you show that for a given basis element M of A and point x in M, there exists a basis element N of B such that x is in N and N is a subset of M. Then you do the same thing in the other direction.

December 9th, 2010, 04:10 PM  #3 
Member Joined: Jan 2008 Posts: 34 Thanks: 0  Re: Open Balls...
I have only one basis, and I need to show that it is indeed a basis... could you be more specific? maybe wrt the objects I mentioned? I'm not very fluent in topology proofs/jargon.


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