I've been studying general topology this term, and there's one basic fact that I don't get. Let $X$ be a topological space, and A some subset.We defined a neighbourhood of $A$ to be all 'points' $x$ such that there's an open set $O$ such that $x$ belongs to $O$ and $O$ is a subset of $A$ (here, no problem). Yet, we defined closure to be all points $x$ such that $X\setminus A$ is not a neighbourhood of $x$.
Here, I have a problem. I would say that not to be a neighbourhood amounts to say that for every open set $U$ (in $X$), either $x$ belongs to $X$, in which case there's some point $y$ that belongs to $X$ and $A$ at the same time (part 1), or $x$ doesn't belong to $U$, in which case $U$ is a subset of $X\setminus A$.(part 2)
But, in other textbooks (like Munkres's General Topology), the closure of $A$ is a set of $x$ such that for every open set $U$ containing $x$, there's a point in intersection between $U$ and $A$.
Here, I'm confused, because it seems to me that these definitions of closure are not equivalent, the second one only having the part 1 of the first definition and not the part 2.
(PS: I've looked at other threads discussing closure, but nobody seems to have the first definition of closure).