The basic idea is correct, except that is insufficient for the proof. For example, let and . Then . If you choose then you may get something like , in which case , which is not disjoint from .
Instead let , and this will suffice for the proof.
Q: Show that a closed ball is always closed in the meteric space?
This is my answer. Is it true?
Let t be any point in this complement of B(a;r). Compute the distance between the center of the closed ball and t; call it d.
By setting epsilon = d/2, the open neighborhood (ball) centered at t with radius epsilon is disjoint from the closed ball B(a;r).
Hence, the complement of the closed ball B(a;r) is open, thus the closed ball is a closed set.
could any one help me in this.
The basic idea is correct, except that is insufficient for the proof. For example, let and . Then . If you choose then you may get something like , in which case , which is not disjoint from .
Instead let , and this will suffice for the proof.