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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.

Community
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GovEcon
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1 Answers1

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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.

dfeuer
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    @Jordan Mahar: The critical thing is that an open neighborhood includes a ball, but may be different. In $\Bbb R, (-1,1)$ is an open ball around $0$ with radius $1$. It is also a neighborhood of $\frac 12$ because it includes an open ball of radius $\frac 12$ (or any smaller positive number) around $\frac 12$. Usually a neighborhood can contain more points than the ball, but the definition you cite seems not to allow that. – Ross Millikan Jun 22 '13 at 03:43
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    Yes, indeed, the definition you cite from Rudin is not currently a common one. It may have been written when the field was younger and the terminology even wilder than now. To add on to @RossMillikan in another way, see http://www.proofwiki.org/wiki/Open_Set_may_not_be_Open_Ball which shows that any metric space with at least three points must have an open set which is not an open ball. – dfeuer Jun 22 '13 at 03:46
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    @RossMillikan and dfeuer. Thank you for your responses. I believe I understand. Essentially an open neighbourhood contains a ball, but may or may not actually be a ball itself. – GovEcon Jun 22 '13 at 03:49
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    @JordanMahar: Exactly. It needs to be an open set that includes a ball around the point in question, but may have anything else included. It still needs to be an open set, so $(0,1) \cup (8,9)$ is an open neighborhood of $\frac 12$ in the usual definition. But you need to pay attention to the definition in each publication-given the definition you cite for a neighborhood, there will be different theorems that can be proved compared to the one dfeuer and I have cited. – Ross Millikan Jun 22 '13 at 04:03
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    But remember that a neighborhood, in most but all texts, need not be an open neighborhood. So you will see references to closed neighborhoods, compact neighborhoods, connected neighborhoods, etc., which in most cases are not assumed to be open. – dfeuer Jun 22 '13 at 04:09
  • @RossMillikan But using definitions of ball and meighborhood from Rudin written in the OP, a ball is a neighborhood and a neighborhood is a ball, right? – Ovi Oct 10 '17 at 13:16