# Empty Set Metric Space

• zooxanthellae
In summary, There is no problem with defining a metric space on the empty set, as all the required properties can still be satisfied. It is not necessary to explicitly state that the set is non-empty in the axioms for a metric space. This convention allows for easier application of theorems, such as the fact that every non-empty subspace of a metric space is also a metric space. A metric space is a function from one set to another, satisfying certain conditions. In the case of the empty set, it can be viewed as a function from the empty set to any other set, vacuously satisfying all the conditions.

#### 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!

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.

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."

OK, thanks to you both.

zooxanthellae said:
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!

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).

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

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.

Deveno said:
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).

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