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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.

Mittens
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Gibson Naegle
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    A neighborhood of a point $x$ is an open set containing $x$. In the case of the complex plane, for example, (arbitrarily small) balls centered at $x$ are neighborhoods of $x$. – RSpeciel May 25 '21 at 22:18
  • Are there any conditions that must be met in this arbitrarily small ball centered at $x$? For example by "arbitrarily small" do you just mean we can pick a $\epsilon > 0$ such that the boundaries are all $\epsilon$ far from $x$ and the boundaries are still in the complex plane? – Gibson Naegle May 25 '21 at 22:22
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    The Wikipedia article for "neighborhood" in mathematics is accurate and comprehensive of the two definitions I've seen. – Mark S. May 25 '21 at 22:31
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    @GibsonNaegle Any open ball centered at $x$ (no matter how large) is a neighborhood of $x$, I was just emphasizing that you can take this ball to be as small as you want and it will still meet the definition of a neighborhood. – RSpeciel May 25 '21 at 22:37
  • Consistent with the above comments, the idea is that if there exists a neighborhood around $z_0 \in \Bbb{C}$ such that a property $P(z)$ holds, then there exists some $r > 0$ such that for all $z$ such that $|z - z_0| < r$, the property $P(z)$ holds. – user2661923 May 25 '21 at 23:03
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    The meanings in real or complex analysis are just special cases of the meaning in topology, using the usual topology on $\mathbb{R}$ or $\mathbb{C}$. – Eric Wofsey May 25 '21 at 23:33

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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 $p$, $p\in U\subseteq V$. … 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.

Mark S.
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