An empty set is closed, open, bounded, convex... All of that is vacuously true.

I wonder which properties are false for empty set?


closed as too broad by Eevee Trainer, Derek Elkins, Shailesh, Lord Shark the Unknown, Lee David Chung Lin May 16 at 4:59

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    $\begingroup$ It's false that the empty set is nonempty. $\endgroup$ – Clayton May 16 at 1:37
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    $\begingroup$ what other properties are you considering? $\endgroup$ – Graham Kemp May 16 at 1:37
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    $\begingroup$ Any statement that starts "for all $x \in X$..." will be true if $X=\emptyset$. Any statement that starts "there exists $x \in X$..." will be false if $X=\emptyset$. $\endgroup$ – kccu May 16 at 1:39
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    $\begingroup$ As an example, the empty set is not a star domain. $\endgroup$ – kccu May 16 at 1:41
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    $\begingroup$ @N.S. In the definition of a topology the empty set is required to be closed and open. And it IS convex. $\endgroup$ – DanielWainfleet May 16 at 2:28

The key distinction has to do with quantifiers: is the property in question universal or existential?

Any property of the form "Every element is [stuff]" is trivially satisfied by the emptyset since the emptyset has no elements which could fail to be [stuff].

Conversely, any property of the form "Some element is [stuff]" is trivially false of the emptyset.

Note that "for all" and "there exists" are dual to each other: the negation of a universal statement is an existential statement, and conversely. So any property referring to elements only will be either trivially true of the emptyset or trivially false of the emptyset.

  • $\begingroup$ Note @kccu's comment to a similar effect. $\endgroup$ – LSpice May 18 at 21:12

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