neighborhood
A neighborhood of a point is, informally, any chunk of the space that surrounds the point with a bit of room to spare on every side. It is the topological version of the calculus phrase “for all x sufficiently close to a” — a neighborhood is precisely the set of points that are, in the relevant sense, near enough.
Precisely, a neighborhood of a point p in a topological space is any set N that contains some open set U with p in U. Many books require N itself to be open and call that an open neighborhood; both conventions are common, so it is worth checking which one an author uses. The collection of all neighborhoods of a point is its neighborhood system.
Neighborhoods turn topology back into local language. Continuity at a point, convergence of a sequence, and limit points can all be phrased neighborhood-by-neighborhood without ever naming the whole topology — for instance, a sequence converges to p exactly when every neighborhood of p contains all but finitely many terms of the sequence.
On the real line, (0.9, 1.2) is an (open) neighborhood of the point 1; so is the larger set [0, 5], which is not itself open but contains the open set (0.9, 1.2) around 1. By contrast [1, 2) is NOT a neighborhood of 1, because no open interval around 1 fits inside it — every open interval around 1 pokes out to the left of 1.
A neighborhood must give the point room on every side; a half-open set fails to be a neighborhood of its endpoint.