Definition: Neighborhood
Let $A$ be an element of a topological space $(X,\mathcal O)$ ($A$ might be a single point of a set of points). A neighborhood $N_A$ is any subset of $X$ containing an open set containing $A$, formally $$N_A\in \{U\subseteq X\mid \exists O\in\mathcal O, A\in O\subseteq U\}.$$
The set of all neighborhoods of $A$ is denoted by $\mathcal N(A).$
Notes
 $X$ is a neighborhood of all of its points.
 Any open set $O\in\mathcal O$ is a neighborhood of all of its points.
 A neighborhood itself does not have to be open (e.g. it is closed but contains an open subset containing $A$, or it is neither open nor closed but contains a subset containing $A$).
Mentioned in:
Axioms: 1
Definitions: 2
Parts: 3
Proofs: 4 5 6 7 8
Propositions: 9 10 11
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
 Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
 Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition