Meromorphic functions on an open set in complex Euclidean space

Fix $n\ge 1$ and read ${\mathbb{C}}^n$ through Complex $m$-space and its real coordinate dictionary. Let $U\subseteq {\mathbb{C}}^n$ be a nonempty open set, and for an open $W\subseteq {\mathbb{C}}^n$ write $\mathcal{O}(W)$ for the set of holomorphic functions $W\to \mathbb{C}$ (Holomorphic functions on an open subset of ${\mathbb{C}}^m$). Open subsets of ${\mathbb{C}}^n$ carry the subspace topology, and components are those of Connected components, quasicomponents, and totally disconnected spaces.

(a) Meromorphic functions. A meromorphic function on $U$ is a function $F\text{\hspace{0.17em}⁣}:D\to \mathbb{C}$ such that

1. $D\subseteq U$ is open and dense in $U$;
2. $F$ is holomorphic on $D$;
3. every point $p\in U$ has a neighbourhood $W\subseteq U$ and holomorphic functions $f,g\in \mathcal{O}(W)$, with $g$ not identically zero on any connected component of $W$, such that $F(z)=\frac{f(z)}{g(z)}\text{whenever }z\in D\cap W\text{ and }g(z)\mathrm{\ne }0.$

The set $D$ is the domain of definition of $F$, briefly its domain. Every $H\in \mathcal{O}(U)$ determines a meromorphic function on $U$ with domain $U$ (take $D=U$ and $g\equiv 1$ in clause 3). A meromorphic representative $F:D\to \mathbb{C}$ with $D\mathrm{\ne }U$ is holomorphic on $U$ when it admits a holomorphic extension as in clause (c); removable omissions from $D$ are therefore allowed.

(b) Restriction. If $F$ is meromorphic on $U$ with domain $D$ and $V\subseteq U$ is open and nonempty, then the restriction $F{\mathrm{\mid }}_V$ is the meromorphic function on $V$ with domain $D\cap V$ and values $(F{\mathrm{\mid }}_V)(z):=F(z)$; the domain $D\cap V$ is dense in $V$ because $D$ is dense in $U$, and clause 3 for $F$ restricts to clause 3 for $F{\mathrm{\mid }}_V$.

(c) Holomorphic on a subset; differences. Let $F$ be meromorphic on $U$ with domain $D$ and let $V\subseteq U$ be open. Then $F$ is holomorphic on $V$ when there is $H\in \mathcal{O}(V)$ with $H(z)=F(z)$ for every $z\in D\cap V$; such an $H$ is called a holomorphic extension of $F$ to $V$. For meromorphic $F,G$ on $U$ with domains $D_F,D_G$ the difference $F-G$ is the function $D_F\cap D_G\to \mathbb{C}$, $z\mapsto F(z)-G(z)$, and for open $V\subseteq U$ the phrase "$F-G$ is holomorphic on $V$" means that $F-G$ admits a holomorphic extension to $V$ in this sense.

(d) Sums with holomorphic functions. If $F$ is meromorphic on $U$ with domain $D$ and $h\in \mathcal{O}(U)$, then $F+h\text{\hspace{0.17em}⁣}:D\to \mathbb{C}$, $z\mapsto F(z)+h(z)$, is meromorphic on $U$: it is holomorphic on $D$, and wherever $F=f\mathrm{/}g$ on $D\cap W$ for a local representation of clause 3 one has $F+h=(f+hg)\mathrm{/}g$ there, while $f+hg$ and $g$ are holomorphic on $W$ and $g$ is not identically zero on any component of $W$ (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

Poles. Let $F$ be meromorphic on $U$ with domain $D$. A point of $U$ is a pole of $F$ when $F$ admits no holomorphic extension to any neighbourhood of the point. The points where $F$ does admit a holomorphic extension form an open subset of $U$: each such extension also works near every point in its domain. Hence the pole set is closed. Every pole lies outside $D$, and $U\setminus D$ is closed with empty interior because $D$ is open and dense, so the pole set also has empty interior. In particular a meromorphic function is never undefined on a nonempty open subset of its ambient set: its domain meets every nonempty open subset of $U$.

Local nature. Clause 3 is a local condition: if every point of $U$ has a neighbourhood to which $F$ restricts as a meromorphic function, then $F$ is meromorphic on $U$. Concretely a ratio of two holomorphic functions is meromorphic on the open set where the denominator does not vanish identically on a component. This is the standard definition in several complex variables: in several variables only the local ratio is available in general.

Choice. This definition quantifies only over points, neighbourhoods and holomorphic functions; it invokes no choice principle, and none of its clauses selects from a family of nonempty sets.