# Empty Set Metric Space

1. Nov 3, 2011

### zooxanthellae

Can we define a metric space $(\emptyset, d)$? The metric is the part that confuses me, since it seems like all of the required properties of d are satisfied since they are "not not satisfied", but I'm not sure.

Thank you!

2. Nov 5, 2011

### Jamma

Yes, I don't see the problem with that. However, I think you would probably want to include "non-empty" in the axioms for a metric space, it's just that you wouldn't usually bother because you don't gain anything of interest by looking at an empty metric space.

3. Nov 5, 2011

### disregardthat

It's a perfectly fine metic space and it's a good thing to keep that convention. So we don't always have to make awkward exceptions to theorems, such as "every non-empty subspace of a metric space is a metric space."

4. Nov 5, 2011

### zooxanthellae

OK, thanks to you both.

5. Nov 6, 2011

### Deveno

how are they not satisfied? every condition for a metric holds for every element of Ø, no matter how we define the metric (although if you must have a definition, use d(x,x) = 0, d(x,y) = 1, for all x,y not in Ø).

6. Nov 6, 2011

### Jamma

... this all comes to vacuous truths again, as discussed elsewhere.

7. Nov 6, 2011

### homeomorphic

What is a metric?

It's a function from X cross X to ℝ, plus some conditions.

If X is the empty set, it's a function from the empty set to ℝ.

What's a function from a set S to a set T?

Formally, it's a subset of S cross T, satisfying some condition.

So, the empty set can be viewed as a function from the empty set to any other set. The empty function. So, that's your metric and it vacuously satisfies all the conditions.

8. Nov 6, 2011

### zooxanthellae

I wrote "not not satisfied", which is a slightly stupider way of saying "vacuously true" as Jamma and homeomorphic have clarified.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook