Definition of neighborhood I am starting to work through Rudin's Principles of Mathematical Analysis.  For $(X, d)$ a metric space and $x \in X$, Rudin defines the neighborhood $N_r(x)$ of $x$ to be the set consisting of all points $y$ such that $d(x,y) < r$.  My question is this: if $A \subset X$ and $a \in A$, then does $N_r(a)$ refer to the same set in both $A$ and $X$ the ambient space?  For example, if $X = \mathbb{R}$ and $A = \mathbb{Q}$, then is it the case that $N_{1/2}(a)$ is always the set $N = \{p \in \mathbb{R} : a - 1/2 < p < a + 1/2\}$, or can $N_{1/2}(a)$ be interpreted as the set $N' = \{q \in \mathbb{Q} : a - 1/2 < q < a + 1/2\}$?  Essentially I am asking if the notion of neighborhood is relative.  This would affect things like the interiority of points in a set.  For instance, $N$ is always contained in $\mathbb{R}$ so that $a$ is always an interior point of $\mathbb{R}$, but $N \not \subset \mathbb{Q}$ so that $a$ is never an interior point of $\mathbb{Q}$.  On the other hand, $N'$ is always contained in both $\mathbb{R}$ and $\mathbb{Q}$ so that $a$ is always an interior point of both of them.  (I guess $N$ is basically a neighborhood formed with respect to the whole space while $N'$ is a neighborhood formed with respect to a specific subspace of the metric.)  Thanks in advance.

Most certainly a relative notion. You define the neighborhood $N_r(x)$ of $x$ in the metric space $X$. So, in particular, a neighborhood in the Euclidean space $\mathbb R$ is very different than a neighborhood in the subspace $\mathbb Q$. The former contains uncountably many elements, while the latter only countably many, and all are rational numbers. 
It should be noted that modern texts refer to $N_r(x)$ as an open ball (and usually denoted by $B_r(x)$), while a neighborhood of $x$ is a term reserved more commonly to one of two things: either an open set containing $x$, or an arbitrary set containing a set of the form $B_r(x)$.
