Thread: metric spaces

Define a metric on a set having more than one element as follows:
For every , d(x,y)={1 if x =/= y, 0 if x = y}
Let in . Give an example that shows the closure of B(b,1) =/= {y in Y : d(b,y)<=1}
NOTE: B(b,1) is an open ball centered at b with radius 1.

Originally Posted by dori1123

Define a metric on a set having more than one element as follows:
For every , d(x,y)={1 if x =/= y, 0 if x = y}
Let in . Give an example that shows the closure of B(b,1) =/= {y in Y : d(b,y)<=1}
NOTE: B(b,1) is an open ball centered at b with radius 1.

By definition .
Now does not intersect .
Thus, is not a boundary point.
Thus, .
Now, .

And we see that .

Hello,

If you want more information, this is called the discrete metric.

Discrete space - Wikipedia, the free encyclopedia