Def. about neighborhood on wiki The def on wiki:
"If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V \subseteq X$ that includes an open set $U$ containing $p$."
And it says: "Note that the neighbourhood $V$ need not be an open set itself."
Later, it says: "A set that is a neighbourhood of each of its points is open."
The claims above seem odd; one says a neighborhood need not be open; the other, be open.
I know the difference is one says a point; the other, a set. Is there any other 
strict explanation?
Thanks

The statement you have doesn't contradict the fact that a neighborhood need not be open, it just happens to be open, given the added condition that it is a neighborhood to each of its points:
Let $V$ be a neighborhood of some point $p$. Suppose $V$ is a neighborhood for each of its points, that means if $x \in V$, then there exists $U$ open with $x \in U \subseteq V$. Conclude that $V$ is open by definition of open set.
