Is a Metric Space with Infinite Distance Totally Bounded?

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Discussion Overview

The discussion revolves around the properties of a metric space, specifically whether a metric space with infinite distance can be considered totally bounded. Participants explore the implications of the definitions of boundedness and total boundedness in the context of a metric space consisting of two points, one of which is infinite.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether a metric space with points X={a,∞} can be totally bounded despite having an infinite distance between the points, noting that for all ε>0, the space can be expressed as the union of balls around each point.
  • Another participant references a proof that states totally bounded implies bounded, highlighting that the proof assumes finite distances between points, which raises questions about the validity of the implication in the case of infinite distances.
  • A third participant asserts that the assumption of finite distances is inherent in the definition of metrics, suggesting that the discussion hinges on this foundational aspect.
  • A fourth participant reiterates that the axioms of metric spaces require distances between any two points to be real and finite, further emphasizing the implications of this definition on the discussion of total boundedness.

Areas of Agreement / Disagreement

Participants express disagreement regarding the implications of total boundedness in the context of infinite distances, with some asserting that the definitions of metric spaces preclude such cases, while others challenge the applicability of the established proofs.

Contextual Notes

The discussion highlights limitations related to the definitions of metrics and the assumptions about distances, particularly in cases involving infinite values. The implications of these definitions on the properties of total boundedness remain unresolved.

johnqwertyful
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It seems strange, but would a metric space consisting of two points, X={a,∞} be totally bounded, but not bounded? because d(a,∞)=∞. But for all ε>0, X=B(ε,a)UB(ε,∞).

It's been proven that totally bounded→bounded, so this is wrong. Why?
 
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johnqwertyful said:
So if we assume that the distance between every two points is finite

This "assumption" is incorporated into the definition of metrics.
 
the axioms for a metric space state that for any two points in the metric space, their distance is a real (and finite) number.
 

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