- #1
CCMarie
- 11
- 1
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.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.If X is a metric space, what is the simplest sufficient condition for d(x,y)<∞, ∀ x, y ∈ X?
For any two points the distance is always finite.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.
I think there are extended metrics that allow for infinite distance, but they are nonstandard.For any two points the distance is always finite.
Interesting question: Do we have a metric on the projective line?I think there are extended metrics that allow for infinite distance, but they are nonstandard.
But these may be different questions: the Real line, nor Euclidean n-space is bounded.Interesting question: Do we have a metric on the projective line?
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).Sure. I was thinking about including infinite points and a non standard metric you mentioned.
What is the "standard" metric in Projective space? Is it the quotient metric?EDIT: Do we even know if the projective space is metrizable?That's why I said projective space. Can we keep the metric via the transition from circle to line, sphere to plane?
BAH. It is a manifold and manifolds are metrizable. With weird exceptions for 1-2 people who allow non-Hausdorff.I have no idea, that's why I asked.
When it comes to topology I need to see. If possible step by step. Those people are notoriously counterintuitive.BAH. It is a manifold and manifolds are metrizable. With weird exceptions for 1-2 people who allow non-Hausdorff.
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 :When it comes to topology I need to see. If possible step by step. Those people are notoriously counterintuitive.