From Wikipedia
If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$, which includes an open set $U$ containing $p$,
Note that the neighbourhood $V$ need not be an open set itself. If $V$ is open it is called an open neighbourhood. Some authors require that neighbourhoods be open, so it is important to note conventions.
Clearly neighbourhood and open neighbourhood are two different concepts. But in the limited number of statements I have seen and can recall, neighbourhood and open neighbourhood can always replace each other without changing the statements from true to false, or from false to true.
So I was wondering if neighbourhood and open neighbourhood can always be exchangeable in statements? If not always, is it most of the cases? What are some statements where exchanging between neighbourhood and open neighbourhood matters? Well, a trivial example is: "an open neighbourhood is a neighbourhood" is true while "a neighbourhood is an open neighbourhood" isn't. But these exemplar statements are not really meaningful.
What is the purpose of distinguishing between neighbourhood and open neighbourhood?
Thanks and regards!