formalize definition of topology

Question

In my studies I used this definition of topology, but I am reading on wikipedia a different definition... I thought to formalize:

Def. let be $A$ a set and $B \in \mathcal{P}(\mathcal{P}(A))$, $(A,B)$ is topology if

• $\emptyset \in B$
• $A \in B$
• $\forall X \subseteq B( \bigcup X \in B)$
• $\forall X_1,X_2,X_3,...,X_n \in B((...((X_1 \cap X_2) \cap X_3) ... \cap X_n) \in B)$

It is correct? Thanks in advance!

Answer 1

Well, it's okay. But here are two points that I'd change:

1. $B$ is the topology. The pair $(A,B)$ is a topological space.

2. It suffices to require that $\forall X,Y\in B(X\cap Y\in B)$.

   As it is written, the last axiom is very hard to understand, both due to excessive parentheses (intersection is associative, so we can remove all of them anyway) and because it's unclear whether or not $n$ is constant. If it isn't, there should be a quantifier on $n$ before the whole thing; if it is then why not pick $n=2$?