What is meant by "neighbourhood?" I was reading about poles of complex functions on Wikipedia and it says "A function $f$ of a complex variable z is meromorphic in the neighbourhood of a point $z_0$ if either $f$ or its reciprocal function $1/f$ is holomorphic in some neighbourhood of $z_0$." The highest math classes I've taken are abstract algebra and ordinary differential equations so I imagine this is well-understood term in higher maths that I haven't reached yet. I had a professor who used the term a lot when talking about differential geometry and also I've seen it frequently when I read about topological spaces; but, articles generally skip over the term with the expectation that the reader understands.
What is rigorously meant by neighbourhood, especially in complex or real analysis (I mostly understand the idea in the context of topology)? Could someone point me to resources that might shed some light on its definition in different contexts?
P.S. I was really unsure of what tags to put and how to ask this question concisely; if anyone has suggestions for edits, I'd be happy to fix it.
 A: In the most general context, "neighbourhood" is defined in the field of topology, where we have a space, and usually designate certain subsets as "open". Quoting Wikipedia:

If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$, $p\in U\subseteq V$.
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Some authors have been known to require neighbourhoods to be open, so it is important to note conventions.

In the special case of a metric space, where the topology comes from a real-valued distance function (e.g. $d(z,w)=|z-w|$ in the complex plane), it follows that (quoting Wikipedia again):

a set $V$ is a neighbourhood of a point $p$ if there exists an open ball with centre $p$ and radius $r>0$, such that $B_r(p)=B(p;r)=\{x\in X\mid d(x,p)<r\}$ is contained in $V$.

To be explicit, this means that, unless a non-standard topology is being used, a neighbourhood of a point $z_0\in\mathbb C$ is any set $V\subseteq \mathbb C$ containing a disk of the form $\{z:|z-z_0|<r\}$ for some positive radius $r$ (possibly very small). As illustrated on Wikipedia, a closed rectangle is not a neighbourhood of its corners or edge points; but it is a neighbourhood of its other points (assuming positive area). Note that "Property $P$ holds on some neighborhood of $z_0$." is logically equivalent to "Property $P$ holds on a disk (of positive radius) centered at $z_0$.".
As an aside, we can actually use neighbourhoods instead of open sets to define a topological space, as mentioned in the definition via neighbourhoods section on Wikipedia.
