Firstly, my definitions, in which this question lies:
Let $U$ be an open set in $\mathbb{R}^{n}$. $f:D\to \mathbb{R}\cup\{-\infty\}$ is subharmonic if
- $f$ is upper semi-continuous in $U$.
- For every open ball $B_{2}(a,r)$ (with $a\in\mathbb{R}^{n}$ y $r\in\mathbb{R}$) such that $\overline{B_{2}(a,r)}\subseteq U$ and for each $g:\overline{B_{2}(a,r)}\to \mathbb{R}\cup\{-\infty\}$ continuous function in $\overline{B_{2}(a,r)}$ and harmonic in $B_{2}(a,r)$, such that $f(x)\leq g(x)$, for all $x\in\partial B_{2}(a,b)$, we have $f(x)\leq g(x)$, for all $x\in B_{2}(a,r)$.
Let $D\subset \mathbb{C}^{n}$ be a domain (open and connected). $f:D\to \mathbb{R}\cup\{-\infty\}$ is plurisubharmonic if:
- $f$ is upper semi-continuous in $D$.
- $\forall a,b\in\mathbb{C}^{n}$, \begin{equation*} \begin{array}{cccc} f_{a,b}:&\mathbb{C}\cong\mathbb{R}^{2} &\longrightarrow& \mathbb{R}\cup \{-\infty\}\\ &z&\longmapsto&f(a+bz) \end{array} \end{equation*} is subharmonic in $\{z\in\mathbb{C}: a+bz\in D\}$.
My question is rather simple... There is a place on the definition of plurisubharmonic function where I have some doubts...
Why is $\{z\in\mathbb{C}: a+bz\in D\}$ an open set in $\mathbb{C}\cong\mathbb{R}^{2}$?
Any hint/ advice is appreciated... I just don't want to work with this concepts until I understand them as accurately as I can. Thanks in advanced!
For fixed $a, b \in \Bbb C^n$ is $$ h: \Bbb C \to \Bbb C^n, h(z) = a+bz $$ a continuous function, and $$ \{z\in\mathbb{C}: a+bz\in D\} = h^{-1}(D) $$ is open as the preimage of an open set under a continuous function.