On a topological space $X$, the class of all neighborhoods of $x\in X$ formalizes the intuitive concept of "closeness" to the point $x$. These sets are usually defined using open sets on $X$. However, one can also define the set of neighborhoods axiomatically and back out the axioms for an open set, and these two definitions will be equivalent. The axioms neighborhoods are give as the following.
$\mathcal{N}(x)$ is a set of neighborhoods of $x\in X$ iff
- If $N\in \mathcal{N}(x)$, then $x\in N$
- If N contains some $M\in \mathcal{N}(x)$, then $N\in \mathcal{N}(x)$
- If $N,M\in\mathcal{N}(x)$, then $N\cap M\in\mathcal{N}(x)$
- If $N\in\mathcal{N}(x)$, then there is some $M\in\mathcal{N}(x)$ such that for each $y\in M$, $N\in\mathcal{N}(y)$.
Informally this is saying that (1) neighborhoods of $x$ contain $x$, (2) supersets of neighborhoods are neighborhoods, (3) intersections of neighborhoods are neighborhoods. (4) is more complicated but it is necessary to ensure that $N(x)$ and $N(y)$ are consistent in terms of "closeness". That is if $x$ is "close" to $y$, then $y$ is "close" to $x$.
My question is about axiom (2). What is the utility of this axiom? It seems counter intuitive to me. That is an $\epsilon$-ball around a point $x$ makes sense as a neighborhood, but why is this ball plus any other set of points also a neighborhood? I can justify this axiom by saying that a neighborhood includes every point "close" to $x$, but not every point of the neighborhood has to be "close" to $x$. However, I am wondering about the necessity of this axiom. Why do we need to include it in the definition of a neighborhood? In other words, if we omit (2), it seems like the concept of neighborhoods is still captured.
Best Answer
We may stick to only open neighborhoods or only open local bases (like open balls). That is, in fact, how I first learnt it. For starters, having (2) is convenient because frequently we want to discuss special kinds of neighborhoods. For example:
Without (2), the statement would become
It doesn't seem like much less work in one sentence, but there are situations when we need to repeatedly refer to the same concept. Without (2), each time we would have needed to say "a neighborhood with compact closure" instead of "a compact neighborhood". The latter expression is much more compact.
To add on that, I personally have not encountered any inconvenience caused by having (2). For example:
Its true with or without (2).