I'm trying to puzzle out a statement given in the Wikipedia article on topological neighborhoods, which uses this definition:
If $X$ is a topological space and $p$ is a point in $X$, a neighborhood of $p$ is a subset $V$ of $X$, which includes an open set $U$ containing $p$.
The picture used to illustrate the corner argument is this:
I guess I can understand that if $p$ pictured is a member of $V$'s boundary set, which is closed (by the definition of boundary sets?), then the rectangle $V$ does not include an open set $U$ containing $p$, but that's somehow unsatisfying. What if $V$ were actually equal to the whole space $X$? What if $V$ were closed and open at the same time?
More simply, why does the neighborhood of $p$ have to be a disk that extends both inside and outside of $V$?
I have essentially no experience in topology, I'm just doing this out of curiosity.