Open Subsets in Metric Space A with Discrete Metric d

[Mr_Physics]

Homework Statement

Let A be a non-empty set and let d be the discrete metric on X. Describe what the open subsets of X, wrt distance look like.

Homework Equations

The Attempt at a Solution

I think that the closed sets are the subsets of A that are the complement of a union of singletons, or in other words, every subset is closed.

But what does that say about the open subsets?

 

[jbunniii]

Are the singletons open sets under this metric?

What can you say about a union of open sets?

 

[Mr_Physics]

I think a union of open sets is open.

Not sure if singletons are open or not.

 

[micromass]

Every subset is closed, you say, that is correct. What does this imply for the open sets? Remember that the open sets are exactly the complements of the closed sets!

 

[jbunniii]

Not sure if singletons are open or not.

Hint: If you take a point x in X, then what points does the open ball of radius 1/2, centered at x, consist of?

 

[Mr_Physics]

An open ball of radius 1/2 I guess would just be a singleton, right?

 

[jbunniii]

An open ball of radius 1/2 I guess would just be a singleton, right?

That's right. Therefore every singleton is an open set - in fact, an open ball.

Now what if you have a union of singletons? Is that an open set?