Definition:Neighborhood (Topology)/Neighborhood defined as Open

Definition
Some authorities define a neighborhood of a set $A$ as what defines as an open neighborhood:
 * $N_A$ is a neighborhood of $A$ $N_A$ is an open set of $T$ which itself contains $A$.

That is, in order to be a neighborhood of $A$ in $T$, $N_A$ must not only be a subset of $T$, but also be an open set of $T$.

However, this treatment is less common, and considered by many to be old-fashioned.

When the term neighborhood is used on this site, it is assumed to be not necessarily open unless so specified.