Q:

is it possible for a set not to contain the empty set?

In topology, I read that sometimes we can add the empty set to a set that is not a topology to make it a topology, I'm ok with this, but my question is, how there could be a set that doesn't contain the empty set; is it not that this is the empty set {} and every set whatever in it, from the empty set itself to infinite sets (or any other arbitrary sets) has at least this empty container i.e. the empty set?

A:

If by "$A$ contains $B$", you mean "$B$ is a subset of $A$", then the answer is no. Because the statement

$\forall x\in\{\}: x\in A$

Is always true, it is always true (no matter what $A$ is), that $\{\}\subseteq A$.
If by "$A$ contains $B$", you mean "$B$ is an element of $A$", then the answer is yes. For example, the empty set is not an element of the empty set. Also, it is not an element of $\mathbb N$, or $\mathbb C$, or $\{1,2,3\}$ or a lot of other sets.

