I am reading Bak and Newman's Complex Analysis, and I am having a difficult time understanding the definition of a simply connected region. Here's the definition:
A region $D$ is simply connected if its complement is “connected within $\epsilon$ to $\infty$.” That is, if for any $z_0 \in D^{c}$ and $\epsilon > 0$, there is a continuous curve $\gamma (t), 0 \leq t < \infty$, such that:
i) $d(\gamma(t), D^c) < \epsilon$ for all $t \geq 0$,
ii) $\gamma(0) = z_0$,
iii) $\lim_{t \rightarrow \infty } \gamma(t) = \infty$.
While I understand that, intuitively, the last two conditions state that $D^c$ is unbounded in the sense that any point in the complement can be "joined to $\infty$" using a line/curve that lies within $D^c$. What about the first condition? Also, it'd be great if someone could motivate this definition as well, without invoking algebraic toploogical notions.
It would be instructive to consider the case of a region $D$ with a $2$-dimensional "hole" in it. Taking a point $z_0$ in this hole, we would not be able to construct a curve which "goes to $\infty $" and which stays within $\epsilon $ of $D^c$. Try drawing a picture... This, intuitively, is because $\gamma (t)$ would have to pass through $D$ to get to $\infty$, thereby creating some separation between $\gamma (t)$ and $D^c$, for certain $t$..