What is the advantage (if any) of neighborhoods which are not open? From wikipedia) If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$, $$p \in U \subseteq V.$$ What is the added advantage of defining the neighborhood to be $V$, which need not be open, as opposed to defining the neighborhood to be the open set $U$ from the start?

The short answer: it's convenient.

E.g. one has the definition of local compactness: $X$ is locally compact if every point has a base of compact neighbourhoods. Or another (non-equivalent in general!) definition of local compactness: every point has a compact neighbourhood.

Or (theorem:) a space is regular iff it every point has a base of closed neighbourhoods.

Such definitions or theorems are easier to state with a general notion of neighbourhood. Also, the set of all neighbourhoods is then a filter while the open neighbourhoods are "just" a filter base, etc. This will only appeal to set theory minded people (like myself), probably. We can still say "open neighbourhood of $x$" for an open set that contains $x$, so it's easy to speciaise to that case as well.

For many definitions (like convergence, continuity) it does not matter whether we state them in terms of open neighbourhoods or general neighbourhoods. S owe can be a bit more general using the general term as well.