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 !

Re: Equality of two balls in a metric space

Quote:

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 $\displaystyle x\ne y$ then $\displaystyle d(x,y)>0$.

Now if $\displaystyle x\ne y$ then let $\displaystyle r=\frac{d(x,y)}{2}>0$.

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

How could they be equal ?

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 ?

Re: Equality of two balls in a metric space

Quote:

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 ?

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.