Homotopy definition on Conway's book

Question

I'm studying Complex Analysis on Conway's book "Functions of one complex variable", and there he defines an Homotopy as follows:

A region for him is an open connected set in $\mathbb{C}$ (considering the usual topology). My question here is if the last condition is really necessary? Isn't it just a consequence of the fact that $\gamma_0$ and $\gamma_1$ are closed curves?

Edit: What the book means when saying that the curves are closed is that $\gamma_0(0)=\gamma_0(1)$ and $\gamma_1(0)=\gamma_1(1)$

Edit 2: I just saw that in the exercises section there is the following problem:

This clarifies my question, thanks to the help everybody!

Answer 1

No it's not. Draw a square, the bottom is $\gamma_0$, the top is $\gamma_1$; this tells you nothing about the left and right sides, you need an extra assumption about them in order to keep a closed curve all along.

Answer 2

The basic intuition about a homotopy between two loops (or, more generally, paths) is that it is a continuous deformation of one loop into another. Think about placing two loops of string to a board that has some nails in it. Perhaps one loop is very simple and just wraps around a particular nail once, while the other loop is much more complicated, and maybe looks like it wraps around several nails. If you can untangle the more complicated string so that it lines up exactly with the more simple string—without lifting it over any nails!—then the two paths represented by the two strings are homotopic.

There are a couple of key points here:

1. You can't ever cut the string or lift it over any nails. This means that if recorded the process, you could stop the video at any time and observe an unbroken loop of string that lies flat on the board.
2. The process happens smoothly through time. The string never teleports from one position to another, for example.
3. The string always has finite length. It cannot be stretched out into something infinitely long (which might somehow allow you unwind it without breaking it).
4. Following the above rules or guidelines, we start the process with the complicated loop, and end the process with the simple loop.

Mathematically, this physical process is encapsulated by the defintion

Let $\gamma_0,\gamma_1 : [0,1] \to G$ be two closed rectifiable curves in a region $G$ [this is essentially point (3), above]; then $\gamma_0$ is homotopic to $\gamma_1$ if there is a function $\Gamma: [0,1]\times [0,1] \to G$ such that

1. $\Gamma(s,0) = \gamma_0(s)$ [at time $t=0$, we start with the more complicated loop] and $\Gamma(s,1) = \gamma_1(s)$ [at time $t=1$, we get the more simple loop; together, these are basically point (4) above], and
2. $\Gamma(0,t) = \Gamma(1,t)$ for all $t\in [0,1]$ [this ensure that we always have a loop, i.e. points (1) and (2) above].

This second part of the definition is quite important, as it makes sure that we never, ever cut the string. Without this assumption, every path in a connected region would be homotopic to every other path (imagine cutting the initial loop, shrinking it to a point, transporting that point to a point on the other loop, growing it back up again, and regluing it).