I read two different definitions of neighborhood:
The definition in Flegg's book "From Geometry to Topology": If $X$ is a set which, together with some topology $T$ on $X$, is a topological space, then any subset $N$ of $X$ is a neighborhood of an element $x∈X$ if it includes an open set containing $x$.
The definition from the textbook by O.Y.Viro et al.: By a neighborhood of a point one means any open set containing this point.
The two definitions are different. And textbook in 2 also points out that, as with definition in 1, analysts and French mathematicians (following N. Bourbaki) prefer a wider notion of neighborhood: they use "neighborhood" for any set containing a neighborhood defined in 2. That means, neighborhood are defined ambiguously in pure topological space.
Why neighborhood is defined differently? Is the ambiguity is allowed in its definition, because "neighborhood", which matters in metric space, doesn't matters anymore in topological space, but rather give its place to "open set"?
