Finite distance between two points

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Discussion Overview

The discussion revolves around the conditions under which the distance between two points in a metric space is finite. Participants explore various aspects of metric spaces, including the implications of compactness, boundedness, and the nature of different metrics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that for a metric space X, a sufficient condition for d(x,y) to be finite for all x, y in X might be that X is compact.
  • Others argue that being bounded could be too obvious or circular as a condition.
  • It is noted that for any two points in a standard metric, the distance is always finite, although some extended metrics may allow for infinite distances.
  • Questions arise regarding the existence of a metric on the projective line and whether projective space can maintain a metric through transitions from circle to line or sphere to plane.
  • Some participants express uncertainty about the standard metric in projective space, questioning if it is the quotient metric and whether projective space is metrizable.
  • A later reply mentions that manifolds are generally metrizable, with exceptions for non-Hausdorff cases, indicating a need for careful consideration in topology.
  • One participant provides a proof of boundedness based on the continuity of the distance function on compact sets.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the conditions for finite distances in metric spaces, particularly concerning compactness, boundedness, and the nature of different metrics. The discussion remains unresolved with no consensus reached.

Contextual Notes

Limitations include the dependence on definitions of metrics and the potential for non-standard metrics to introduce complexity. The discussion also highlights unresolved questions about the metrization of projective space and the implications of non-Hausdorff manifolds.

CCMarie
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If X is a metric space, what is the simplest sufficient condition for d(x,y)<∞, ∀ x, y ∈ X?
 
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CCMarie said:
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!
 
CCMarie said:
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.
 
WWGD said:
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.
 
fresh_42 said:
For any two points the distance is always finite.
fresh_42 said:
For any two points the distance is always finite.
I think there are extended metrics that allow for infinite distance, but they are nonstandard.
 
WWGD said:
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?
 
fresh_42 said:
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.
 
Sure. I was thinking about including infinite points and a non standard metric you mentioned.
 
  • #10
fresh_42 said:
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).
 
  • #11
That's why I said projective space. Can we keep the metric via the transition from circle to line, sphere to plane?
 
  • #12
fresh_42 said:
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?
 
  • #13
I have no idea, that's why I asked.
 
  • #14
fresh_42 said:
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.
 
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  • #15
WWGD said:
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.
 
  • #16
fresh_42 said:
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.
 
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