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Thank you!

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- Thread starter zooxanthellae
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- #1

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Thank you!

- #2

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- #3

disregardthat

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- #4

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OK, thanks to you both.

- #5

Deveno

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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).

- #6

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... this all comes to vacuous truths again, as discussed elsewhere.

- #7

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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.

- #8

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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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