Is it posssible for b[x:r) and b[y;s) to be equal with x not equal to y and r not equal to s ?
I know it is possible,for instance if we consider a non empty set X with the discrete metric, then for each x in X the balls b[x;r) for r in (0,1] are equal to the singleton set {x}. Also the balls b[x;r) for r in (1,infinity) are equal to X for all x in X.
What is the idea behind two balls with different radii and centre's being equal ?
What i don't understand is, that even in the above example, in what sense are the two balls equal ?
What is the meaning of equality of two balls in a metric space ?
In this example one ball has only singleton element {x} and the other one is the whole metric space X then how are they equal ?
I am a little confused !
Please read this page on Hausdorff spaces.
Every metric space is a Hausdorff space. So points are separated.
Balls are determined by first a point and a positive real number.
This if balls are equal the the centers are the same.