Homotopy of curves

There are two notions of homotopy
Definition
Let ${\gamma }_0,{\gamma }_1:[0,1]\to A\subset \mathbb{C}$ be two curves such that
$z_0={\gamma }_0(0)={\gamma }_1(0)$ and $z_1={\gamma }_0(1)={\gamma }_1(1)$.

We say that ${\gamma }_0$ and ${\gamma }_1$ are homotopic with fixed endpoints in $A$ if there exists a continuous function
$H:[0,1]\times [0,1]\to A$ such that:

1. $H(0,t)={\gamma }_0(t)$ for all $t\in [0,1]$,
2. $H(1,t)={\gamma }_1(t)$ for all $t\in [0,1]$,
3. $H(s,0)=z_0$ for all $s\in [0,1]$,
4. $H(s,1)=z_1$ for all $s\in [0,1]$.

Definition
Let ${\gamma }_0,{\gamma }_1:[0,1]\to A\subset \mathbb{C}$ be two closed curves (i.e., ${\gamma }_0(0)={\gamma }_0(1)$ and ${\gamma }_1(0)={\gamma }_1(1)$).
We say that ${\gamma }_0$ and ${\gamma }_1$ are homotopic as closed curves in $A$ if there exists a continuous function
$H:[0,1]\times [0,1]\to A$ such that:

1. $H(0,t)={\gamma }_0(t)$ for all $t\in [0,1]$,
2. $H(1,t)={\gamma }_1(t)$ for all $t\in [0,1]$,
3. $H(s,0)=H(s,1)$ for all $s\in [0,1]$.

From here we define the notion of simply connected set.

Related: fundamental group.