[Math] the difference between a ball and a neighbourhood

general-topologymetric-spaces

I am presently reading chapter two of Rudin, Principles of Mathematical Analysis (ed. 3). He provides the following definitions:

Definition: If $\boldsymbol{x} \in \mathbb{R} ^ k$ and $r > 0$, the open ball $B$ with center at $\boldsymbol{x}$ and radius $r$ is defined to be the set of all $\boldsymbol{y} \in \mathbb{R} ^ k$ such that $| \boldsymbol{y} – \boldsymbol{x} | < r.$

Definition: A neighbourhood of a point $p$ is a set $N_r(p)$ consist of all points $q$ such that $d(p,q) < r$. The number $r$ is called the radius of $N_r(p)$.

What I have been attempting to figure out is what the difference between these two definitions are. Ilya, in the following question provides the following description –

"The neighborhood of a point $x\in \Bbb R$ is any subset $N_x\subseteq \Bbb R$ which contains some ball $B(x,r)$ around the point $x$. Note that in general one does not ask neighborhood to be open sets, but it depends on the author of a textbook you have in hands."

In Kaplansky, Set Theory and Metric Spaces he presents an example where by setting the distrance function $d(p,q) = |a-b|$ then we obtain a metric space. Then combining this statement and the part of Ilya's answer that a neighbourhood contains some ball, a ball is like a special case of a neighbourhood.

The following two theorems seem to give support for this case:

Theorem $27$ from Kaplansky: Any open ball in a metric space is an open set.

Theorem $2.19$ from Rudin: Every neighbourhood is an open set.

So essentially they are both open sets, however, the neighbourhood has a more general distance function.

I would appreciate some clarification on this matter.

Best Answer

As you have seen, different texts define their terms somewhat differently, but the most common definitions are as follows:

  • If $(X,d)$ is a metric space (or a pseudometric space), then the open ball of radius $r > 0$ about the point $x \in X$ is the set of all $y \in X$ such that $d(x, y) < r$.
  • If $(X,d)$ is a metric space (or a pseudometric space), then a set $U \subseteq X$ is open iff for each $x \in U$ there is an $r > 0$ such that the open ball about $x$ of radius $r$ is a subset of $U$.
  • An open neighborhood of a point $x$ in a metric space (or, in fact, any topological space) is any open set containing $x$.
  • A neighborhood of a point $x$ in a metric space (or any topological space) is any subset of the space including, as a subset, an open neighborhood of $x$.

Beware: the notations used for open balls vary radically among texts, with almost all imaginable permutations of where the point goes, where the radius goes, and (in some cases) where the name of the metric goes.

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