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CCMarie
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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 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.CCMarie said: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.WWGD said:For X, it is maybe that X is compact. Being bounded may be too obvious/circular.
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.fresh_42 said:For any two points the distance is always finite.
Interesting question: Do we have a metric on the projective line?WWGD said: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.fresh_42 said: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).fresh_42 said: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?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?
BAH. It is a manifold and manifolds are metrizable. With weird exceptions for 1-2 people who allow non-Hausdorff.fresh_42 said: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.WWGD said: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 :fresh_42 said:When it comes to topology I need to see. If possible step by step. Those people are notoriously counterintuitive.
The finite distance between two points is the length of the shortest path connecting the two points. It is a measure of the physical distance between the two points in a straight line.
The finite distance between two points can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the other two sides.
The units of measurement for finite distance between two points can vary depending on the system of measurement being used. In the metric system, it is typically measured in meters, while in the imperial system, it is measured in feet or miles.
No, the finite distance between two points cannot be negative. It is always a positive value, as it represents the physical distance between the two points and cannot be in the opposite direction.
The finite distance between two points is closely related to other mathematical concepts such as vectors, displacement, and magnitude. It is also an important concept in geometry and can be used to calculate the length of curves and surfaces.