Originally Posted by

**alexandros** I presented a professor with the following proof:

Prove that the empty set is closed.

Proof :

By definition : $\displaystyle \emptyset$ is closed <=> cl$\displaystyle \emptyset\subseteq\emptyset$ <=> ($\displaystyle x\in$ cl $\displaystyle \emptyset$=>$\displaystyle x\in\emptyset$)

cl $\displaystyle \emptyset$ is the closure of the empty set,and

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

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

But , B(x,r)$\displaystyle \cap\emptyset =\emptyset$ => (B(x,r)$\displaystyle \cap\emptyset =\emptyset$ or $\displaystyle x\in\emptyset$) <=>

(B(x,r)$\displaystyle \cap\emptyset\neq\emptyset$ =>$\displaystyle x\in\emptyset$)

And using (1) we get : $\displaystyle x\in\emptyset$

Thus ,we have proved:

($\displaystyle x\in$ cl $\displaystyle \emptyset$=>$\displaystyle x\in\emptyset$)

And the empty set is closed.

The professor did not accept the proof as correct .

Do you agree with him??