How is the word "contains" defined in set theory? (In relation with neighborhoods in topology). From Wiki:

Some basic sets of central importance are the empty set (the unique set containing no elements)

Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is contained in $X$, then we write $a \in X$.
However, from this site:

A neighborhood of $z$, which can be denoted $N_z$, is any subset of $S$ containing an open set of [the topological space] $T=(S,\tau)$ which itself contains $z$.
  That is:
  $$
\exists U \in \tau:z \in U \subseteq N_z \subseteq S
$$

But shouldn't it actually be:
$$
\exists U \in \tau:z \in U \ni N_z \subseteq S
$$
because it says that $N_z$ contains an open set $U$? This is also consistent with the statement that $U$ contains $z$ which they write as $z \in U$.
P.S. I'm only interested in set theory from an intuitive point of view and not from an axiomatic point of view.
 A: A useful convention, proposed by Paul Halmos (as far as I know), but unfortunately far from universally accepted, is to use "contains" for  members and "includes" for subsets.  Consider this answer propaganda for that convention.
A: My impression is that most people use the expression "A is contained in B" for "A is a subset of B" and not for "A is an element of B".
A: They're two different uses of the term 'contains'.  While the English the meanings in each is the same, the first example you give refers to the membership relation $x\in S$, which is to say that $x$ is an element of $S$, while the second example you give refers to the subset relation $A\subset B$, which is to say that every element of $A$ is an element of $B$.
This is one of those times when the language commonly used is ambiguous; context is usually enough to differentiate which relation is actually being referred to.  A typical rule of thumb is that if an element of a space is being referred to, then contains refers to the membership relation, while if a set of elements of the space is being referred to, then it refers to the subset relation.  Important to look out for, especially since you're dealing with topology, is that when a topology is being referenced, this rule of thumb does not work as well; the elements of a topology are subsets of the space.  This is why it is important to take into consideration by what is being contained something, as well as what is containing that thing.
A: It would be as defined in the book, for instance a set "contains" elements so we say $x\in U$ but if we have  a set of sets, for instance the power set $2^S$, then the power set contains sets so $U\in 2^S$
