[talanum1 (OP)]

I have read that a sequence converges to a if a is an element of all U an element of O, in a topological space (S, O). Since the empty set must be in O there will never be such a. Does the empty set contain 0?

[Bored chemist]

Does the empty set contain 0?

No, it's empty.

[Eternal Student]

Hi.

   I'm not sure what it was that you read there @talanum1.   It might have got a bit muddled up (maybe some symbols didn't display properly?)   However, I'd go along with Bored Chemist as regards the question in the title.

Best Wishes.

[Bored chemist]

"Is the Empty Set ={0}?"

No
It is not {0}.
The empty set is {}.

It is the set of current kings of the USA
It is the set of carrots on Mars

The number of things in the empty set is zero.
But zero is not in the empty set.

The set of (English names of)  months of the year has 12 members.
But 12 is not a month.