# Holomorphic functions on an open set but not a domain

Let me lay out a definition first.

Definition: A non-empty open connected subset of $$\mathbb{C}$$ is called a domain.

I am currently self-studying Complex Analysis and have been referring to multiple books. Right now, I am at the point of trying to understand the definition of holomorphic functions.

What got me confused is that in some books, they define holomorphic functions on a domain (cf. Lecture Notes by Ivan F. Wilde) while on some books, they define holomorphic functions on open subsets of $$\mathbb{C}$$ (cf. Stein-Shakarchi Complex Analysis, Priestley's Intro to Complex Analysis, Lang's Complex Analysis).

So, my question is, why is this so? Would there be major consequences from this slight difference of definition? Thanks in advance.

Yes, there are big differences. For example if $$U$$ and $$V$$ are two disjoint open sets then $$f(z)=1$$ for $$z \in U$$ and $$f(z)=0$$ for $$z \in V$$ gives a holomorphic function on $$\Omega =U \cup V$$. Its zeros have a limit point but the function is not identically $$0$$. This cannot happen when $$\Omega$$ is a domain. Any holomorphic function whose zeros have a limti point is identically $$0$$ in this case.