"Neighborhood" is a word with many different levels of meaning in mathematics.

One of the most general concepts of a neighborhood of a point (also called an epsilon-neighborhood or infinitesimal open set) is the set of points inside an -ball with center and radius . A set containing an open neighborhood is also called a neighborhood.

The graph neighborhood of a vertex in a graph is the set of all the
vertices adjacent to
generally *including* itself. More generally, the th neighborhood of is the set of all vertices that lie at the distance from . The subgraph induced by the
neighborhood of a graph from vertex (again, most commonly including itself) is called the neighborhood
graph (or sometimes "ego graph" in more recent literature).