Defining an Empty Set Metric Space: Understanding Properties of d

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Discussion Overview

The discussion revolves around the definition and properties of a metric space when the set is empty, specifically examining whether a metric space can be defined as (\emptyset, d) and the implications of such a definition.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants propose that defining a metric space with an empty set is valid, arguing that the properties of the metric d are satisfied in a vacuous manner.
  • Others suggest that including "non-empty" in the axioms for a metric space is preferable, as an empty metric space does not yield interesting results.
  • A participant mentions that the convention of allowing an empty metric space helps avoid awkward exceptions in theorems related to metric spaces.
  • There is a discussion about the nature of a metric as a function from a set to the real numbers, with the empty set being viewed as a function that vacuously satisfies the required conditions.
  • One participant clarifies that their earlier statement about properties being "not not satisfied" refers to the concept of vacuous truths.

Areas of Agreement / Disagreement

Participants express differing views on the validity and utility of defining a metric space with an empty set. While some support the idea, others question its relevance and suggest that it may not be beneficial to include in discussions of metric spaces.

Contextual Notes

The discussion includes considerations of vacuous truths and the formal definition of functions, highlighting the nuances in defining metrics for empty sets.

zooxanthellae
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Can we define a metric space [itex](\emptyset, d)[/itex]? The metric is the part that confuses me, since it seems like all of the required properties of d are satisfied since they are "not not satisfied", but I'm not sure.

Thank you!
 
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Yes, I don't see the problem with that. However, I think you would probably want to include "non-empty" in the axioms for a metric space, it's just that you wouldn't usually bother because you don't gain anything of interest by looking at an empty metric space.
 
It's a perfectly fine metic space and it's a good thing to keep that convention. So we don't always have to make awkward exceptions to theorems, such as "every non-empty subspace of a metric space is a metric space."
 
OK, thanks to you both.
 
zooxanthellae said:
Can we define a metric space [itex](\emptyset, d)[/itex]? The metric is the part that confuses me, since it seems like all of the required properties of d are satisfied since they are "not not satisfied", but I'm not sure.

Thank you!

how are they not satisfied? every condition for a metric holds for every element of Ø, no matter how we define the metric (although if you must have a definition, use d(x,x) = 0, d(x,y) = 1, for all x,y not in Ø).
 
... this all comes to vacuous truths again, as discussed elsewhere.
 
What is a metric?

It's a function from X cross X to ℝ, plus some conditions.

If X is the empty set, it's a function from the empty set to ℝ.

What's a function from a set S to a set T?

Formally, it's a subset of S cross T, satisfying some condition.

So, the empty set can be viewed as a function from the empty set to any other set. The empty function. So, that's your metric and it vacuously satisfies all the conditions.
 
Deveno said:
how are they not satisfied? every condition for a metric holds for every element of Ø, no matter how we define the metric (although if you must have a definition, use d(x,x) = 0, d(x,y) = 1, for all x,y not in Ø).

I wrote "not not satisfied", which is a slightly stupider way of saying "vacuously true" as Jamma and homeomorphic have clarified.
 

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