Consider the definition of a line segment in the complex plane: Given $a, b \in \textbf{C}$, let $L(a, b)$ denote the line segment from $a$ to $b$
$$L(a, b) = \{a + t(b - a)| \ 1 \geq \ t \geq \ 0 \ \}.$$
Why is the interior of a line in the complex plane empty?
We have $|e^{i\theta}-1|\leq|\theta|$ for real number $\theta$.
Given $a+t_{0}(b-a)\in L(a,b)$ and an $\epsilon>0$, we let $|\theta|>0$ small enough such that $|(a+t_{0}(b-a))e^{i\theta}-(a+t_{0}(b-a))|=|a+t_{0}(b-a)||e^{i\theta}-1|\leq|a+t_{0}(b-a)||\theta|<\epsilon$, then $(a+t_{0}(b-a))e^{i\theta}\in B(a+t_{0}(b-a),\epsilon)$, but $(a+t_{0}(b-a))e^{i\theta}\notin L(a,b)$.