neighborhood (informal)

A neighbourhood of a point is just ‘the region close to it’ — everywhere within walking distance of your house, say. In analysis it captures the idea of nearness without committing to a single radius: any small enough zone around the point counts.

Informally, a neighbourhood of a point a on the real line is a set containing all points within some positive distance of a — for instance the open interval (a - delta, a + delta) for some delta > 0. Statements about limits and continuity are really statements about what happens on neighbourhoods: ‘f(x) is close to L for x near a’ means f maps a small neighbourhood of a into a small neighbourhood of L.

The word ‘near’ in analysis almost always unpacks into ‘in some neighbourhood’. This informal picture is later made fully precise: in a metric space a neighbourhood is built from open balls, and in a general topological space it is axiomatised directly via open sets. The honest caveat: which sets count as neighbourhoods depends on the notion of distance or topology in force — on the real line it is the familiar intervals, but the same word covers richer spaces where ‘close’ behaves differently.