High School Finite distance between two points

[CCMarie]

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

 

[fresh_42]

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.

 

[CCMarie]

Yes, it is the metric. And thank you!

 

[WWGD]

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]

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]

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]

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]

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]

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

 

[WWGD]

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]

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

 

[WWGD]

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]

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

 

[WWGD]

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]

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]

When it comes to topology I need to see. If possible step by step. Those people are notoriously counterintuitive.

This is the standard case, don't know if there are many others "non-trivially" different: https://en.wikipedia.org/wiki/Non-Hausdorff_manifold#Line_with_two_origins :

EDIT: Proof of boundedness: d(x,y) is continuous on YxY for Y compact. Then d: YxY -->R is a continuous function on the compact set YxY, and so it is bounded.