0
$\begingroup$

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

I wonder which properties are false for empty set?

$\endgroup$

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

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 3
    $\begingroup$ It's false that the empty set is nonempty. $\endgroup$ – Clayton May 16 at 1:37
  • 2
    $\begingroup$ what other properties are you considering? $\endgroup$ – Graham Kemp May 16 at 1:37
  • 7
    $\begingroup$ 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
  • 2
    $\begingroup$ As an example, the empty set is not a star domain. $\endgroup$ – kccu May 16 at 1:41
  • 1
    $\begingroup$ @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
4
$\begingroup$

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.

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.