disputed proof

[triclino]

I presented a professor with the following proof:

 

Prove that the empty set is closed.

 

Proof :

 

By definition : [math]\emptyset[/math] is closed <=> cl[math]\emptyset\subseteq\emptyset[/math] <=> ([math]x\in[/math] cl [math]\emptyset[/math]=>[math]x\in\emptyset[/math])

 

cl [math]\emptyset[/math] is the closure of the empty set,and

B(x,r) is a ball of radius r round x

 

 

But ,by definition again [math]x\in[/math]cl [math]\emptyset[/math] <=> for all r>0 ,B(x,r)[math]\cap\emptyset\neq\emptyset[/math]....................................................................1

 

But , B(x,r)[math]\cap\emptyset =\emptyset[/math] => (B(x,r)[math]\cap\emptyset =\emptyset[/math] or [math]x\in\emptyset[/math]) <=>

 

(B(x,r)[math]\cap\emptyset\neq\emptyset[/math] =>[math]x\in\emptyset[/math])

 

And using (1) we get : [math]x\in\emptyset[/math]

 

Thus ,we have proved:

 

([math]x\in[/math] cl [math]\emptyset[/math]=>[math]x\in\emptyset[/math])

 

And the empty set is closed.

 

The professor did not accept the proof as correct .

 

Do you agree with him??