\begin{definition}[Definition:Neighborhood (Topology)/Neighborhood defined as Open]
Some authorities define a '''neighborhood''' of a set $A$ as what {{ProofWiki}} defines as an '''open neighborhood''':
:$N_A$ is a '''neighborhood of $A$''' {{iff}} $N_A$ 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.
\end{definition}