Strange definition in topology

Question

I am reading a collection of notes from an introductory course on topology. I came across this seemingly bizarre definition and was wondering if somebody could explain the usefulness of it. It seems entirely arbitrary to me.

"If X is a topological space and S is a subset of X then a neighbourhood of S is a set V, which contains an open set U containing S i.e. $S \subset U \subset V \subset X.$"

A lot of these topological definitions just appear to be going down some sort of rabbit hole and there is no hint as to why.

Answer 1

Given a point $x$, it is often necessary to talk about sets which contain all points sufficiently close to $x$. Thus the concept of a neighborhood. "Sufficiently close" is exactly what the concept of open set was created to describe, so this goes from the heuristic

$V$ is a neighborhood of $x$ if it contains every point sufficiently close to $x$.

to the exact

$V$ is a neighborhood of $x$ if it contains an open set about $x$.

or more symbolically

$V$ is a neighborhood of $x$ if there is some open $U$ with $x \in U \subseteq V$.

Your definition is for neighborhoods of sets instead of just points, but is equivalent to just requiring $V$ to be a neighborhood of every point in $S$.

So why not just stick with open neighborhoods, as is quite common in English mathematics, and as William Elliot apparently supports? Because there are a lot of situations where one has to talk about sets that contain every sufficiently close to $x$, but are not necessarily open themselves. While I am writing this, Moishe Cohen has provided one example. Another is the definition of "locally compact". Which do you prefer?

$S$ is locally compact if for every $x \in S$ there is a compact set $V$ and an open set $U$ with $x\in U \subseteq V$.

or

$S$ is locally compact if every $x \in S$ has a compact neighborhood.

The second definition only works when neighborhoods need not be open.

Like all terminology, "neighborhood" is just a shorthand for a useful concept, so that one does not need to specify it in detail every time it is used. Since Moishe has already brought Bourbaki up, I'll mention that Bourbaki's definition of "1" involves an estimated 24000 symbols (if I recall - it has been a very long time since I read it). One would hardly want to repeat that every time you count.

In most cases, it is sufficient to use open neighborhoods, and there are many occasions where open neighbhorhoods are required. For this reason, many authors simply choose to define neighborhoods to be open so that they don't have to say "open" each time, preferring to say "set containing a neighborhood" for those fewer occasions where having the neighborhood be open gets in the way. But this is a matter of preference, and I personally find it more convenient to use "open neighborhood" when open is necessary, and allow neighborhoods to not be open.