# Thread: Discrete metrics, open and closed sets

1. ## Discrete metrics, open and closed sets

Let X be set donoted by the discrete metrics
d(x; y) =(0 if x = y;
1 if x not equal y:
(a) Show that any sub set Y of X is open in X
(b) Show that any sub set Y of X is closed in X

2. a) Pick an $\displaystyle x \in Y$, what is the ball of radius 1/2 around x?
b) Apply a) to the complement of Y

3. Originally Posted by hebby
Let X be set donoted by the discrete metrics
d(x; y) =(0 if x = y;
1 if x not equal y:
(a) Show that any sub set Y of X is open in X
(b) Show that any sub set Y of X is closed in X
Originally Posted by Focus
a) Pick an $\displaystyle x \in Y$, what is the ball of radius 1/2 around x?
b) Apply a) to the complement of Y
More generally, when you have a metric fixed like this you will always be able to find a neighborhood that contains no points. How that affects openeness and closedness is up to you to figure out.

4. so what would be the steps to ans this question?

As I tried to form an answer with the union of balls B(x,r), as in Y of X we have Y= Union (Y) is open for y in Y....ie union of open sets is open therefore y is open.

Then for b) I tried let Y be a subset of X, the X\Y is open and Y is closed....but i need to explain more....please help !

5. Originally Posted by hebby
so what would be the steps to ans this question?
What do you think for the open portion? For the closed portion consider the neighborhood of any point of radius $\displaystyle \frac{1}{2}$. What is significant about it?

6. i wrote for a)

A subset Y of X is open, if and only if Y is an union of open balls....now how would I prove this?

7. Originally Posted by hebby
i wrote for a)

A subset Y of X is open, if and only if Y is an union of open balls....now how would I prove this?
As I said before, what is the open ball of radius 1/2 around a point x i.e. $\displaystyle \{y:d(x,y)<1/2\}$? Without first answering this you cannot do the question.