Proving that something is a set I'm doing problems from my Topology notes. One of them states the following:
Let $A \times B$ (the cartesian product of $A$ and $B$), with $A$ and $B$ sets. Prove that is a set.
How do we prove that something is a set? Not this case in particular, but every 'set' of things.
Is there any axioms that a set have to verify to be a set?
Thanks!
 A: To elaborate a little on the other answer, from the axiom of specification you obtain the empty set; then by the axiom of power set you obtain the set $2 = \{\{\emptyset\},\emptyset\}$. Then, by replacement you obtain the set $\{A,B\}$. By the axiom of union you obtain $A \cup B$. By the axiom of power set you obtain $\mathcal P(\mathcal P(A \cup B))$. Then, from this set you use specification to obtain all elements of the form $\{\{a,b\},\{a\}\}$, where $a \in A$ and $b \in B$. This is then the cartesian product, where $\{\{a,b\},\{a\}\}$ is chosen to represent $(a,b)$.
Note, however, that this is axiomatic set theory, which (at this level) has nothing to do with topology. Perhaps it is a mistake in the notes, or you misread, or the notes discuss an "informal" notion of set (something like, "a well-defined collection of elements") and they want you to convince yourself that this is one. (But that would also be weird, because typically a topology course only comes along when you are already quite familiar with the cartesian product.)
A: First, $\lbrace A, B \rbrace$ exists by the axiom of pairing, so $A\cup B = \bigcup \lbrace A, B \rbrace$ exists by the axiom of union.
The ordered pair $\langle a, b \rangle$ is defined to be $\lbrace \lbrace a \rbrace, \lbrace a, b \rbrace  \rbrace.$
You can see that if $a\in A$ and $b \in B,$ then $\lbrace a \rbrace$ and $\lbrace a, b \rbrace$ are both subsets of $A\cup B,$ so they're both members of $\scr{P}(A \cup B).$  It follows that $\langle a, b \rangle$ is a subset of $\scr{P}\mathrm{(A \cup B),}$ so each $\langle a, b \rangle$ is a member of $\scr{P}(\scr{P}\mathrm{(A \cup B)).}$
Also, you can see that a set $x$ is an ordered pair iff
\begin{align}(\exists a)(\exists b)(\exists c)(\exists d)\Big(&(\forall y)(y \in x \iff (y=c \lor y=d)) \;\land
\\&(\forall y)(y \in c \iff (y=a)) \;\land
\\&(\forall y)(y \in d \iff (y=a \lor y=b))
\\&\!\!\Big).
\end{align}
It follows that we can write
$$A\times B=\lbrace x \in \scr{P}(\scr{P}\mathrm{(A \cup B)) \mid x\text{ is an ordered pair}\rbrace.}$$
So using the axioms of pairing, union, power set, and separation, we've shown that $A\times B$ is a set if $A$ and $B$ are.
A: I guess you are talking about Zermelo-Frankel set theory.

Is there any axioms that a set have to verify to be a set?

No, a set is a primitive notion in ZF, meaning it is undefined.
A single look at the relevant wiki page answers the question about the Cartesian product of sets:
(Speaking about the Kuratowski definition or ordered pairs:)

Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, and specification. 

That is, it is a set by virtue of being constructed out of sets in a way permitted by the axioms of ZF set theory. I leave it to you as an exercise of how to apply these specific four axioms to Kuratowski's definition to build the Cartesian product. Drop me a line in the comments if you need a nudge at some point.
