# Finite distance between two points

• B
If X is a metric space, what is the simplest sufficient condition for d(x,y)<∞, ∀ x, y ∈ X?

fresh_42
Mentor
If X is a metric space, what is the simplest sufficient condition for d(x,y)<∞, ∀ x, y ∈ X?
What is ##d(.,.)##? If it is the metrc, then this is always the case.

Yes, it is the metric. And thank you!

WWGD
Gold Member
If X is a metric space, what is the simplest sufficient condition for d(x,y)<∞, ∀ x, y ∈ X?
For X, it is maybe that X is compact. Being bounded may be too obvious/circular.

fresh_42
Mentor
For X, it is maybe that X is compact. Being bounded may be too obvious/circular.
For any two points the distance is always finite.

WWGD
Gold Member
For any two points the distance is always finite.
For any two points the distance is always finite.
I think there are extended metrics that allow for infinite distance, but they are nonstandard.

fresh_42
Mentor
I think there are extended metrics that allow for infinite distance, but they are nonstandard.
Interesting question: Do we have a metric on the projective line?

WWGD
Gold Member
Interesting question: Do we have a metric on the projective line?
But these may be different questions: the Real line, nor Euclidean n-space is bounded.

fresh_42
Mentor
Sure. I was thinking about including infinite points and a non standard metric you mentioned.

WWGD
Gold Member
Sure. I was thinking about including infinite points and a non standard metric you mentioned.
But even with the standard metric, there is no Real M with d(x,y)<M for all x,y in ##\mathbb R^n ##Edit: Just choose the pair M,0 ( or generalized to n dimensions) for d(x,y).

fresh_42
Mentor
That's why I said projective space. Can we keep the metric via the transition from circle to line, sphere to plane?

WWGD
Gold Member
That's why I said projective space. Can we keep the metric via the transition from circle to line, sphere to plane?
What is the "standard" metric in Projective space? Is it the quotient metric?EDIT: Do we even know if the projective space is metrizable?

fresh_42
Mentor
I have no idea, that's why I asked.

WWGD
Gold Member
I have no idea, that's why I asked.
BAH. It is a manifold and manifolds are metrizable. With weird exceptions for 1-2 people who allow non-Hausdorff.

fresh_42
fresh_42
Mentor
BAH. It is a manifold and manifolds are metrizable. With weird exceptions for 1-2 people who allow non-Hausdorff.
When it comes to topology I need to see. If possible step by step. Those people are notoriously counterintuitive.

WWGD