Thread: Equality of two balls in a metric space

1. Equality of two balls in a metric space

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 !

2. Re: Equality of two balls in a metric space

Originally Posted by mrmaaza123
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 ?
Those of trained in the tradition of R L Moore are distrustful of empty point sets.
One of the most basic properties of metric is: if $x\ne y$ then $d(x,y)>0$.
Now if $x\ne y$ then let $r=\frac{d(x,y)}{2}>0$.

Then it should be very clear that $\mathfrak{B}\left( {x;r} \right) \cap \mathfrak{B}\left( {y;r} \right) = \emptyset$.

How could they be equal ?

3. Re: Equality of two balls in a metric space

The interval in which r lies is changing so isn't "r" changing ? How can we take it to be the same for both the cases ?

4. Re: Equality of two balls in a metric space

Originally Posted by mrmaaza123
The interval in which r lies is changing so isn't "r" changing ? How can we take it to be the same for both the cases ?