Definition of neighbourhood in topology

Question

The definition of neighbourhood:

If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$, which includes an open set $U$ containing $p$.

Does this say in other words that, a neighbourhood is any open set that contains $p$? Why it has to be for a set $S$, $S\subset U\subset V$ and not $S\subset V$?

Answer 1

Usually "neighborhood" (please excuse my American spelling) is allowed to be either open or not open. So it's not enough to say $p \in V$, because then $V$ could be the set consisting only of the point $p$, which wouldn't capture the sense of the word "neighborhood" as a region of points close to $p$. Therefore we require $p \in U \subset V$ where $U$ needs to be open, and $V$ can be open or closed.

Often it won't make much of a difference, and you can just consider the open set $U$ directly.

See the question here, and particularly Pedro's answer, for some further insight.

Answer 2

What it means is, if $U$ is an open set and $p \in U$ then if a subset $V$ of $X$ contains $U$, $V$ is a neighbourhood of $p$. Choose $p=0$, according to the definition, a subset $V\subset R$ like $V=[-0.5,0.5]$, includes the open set $(-0.5, 0.5)$ which contains $p$ ($p$ belongs to the open set $(-0.5,0.5)$). So $V$ is a neighbourhood of $p$.