Descriptive explanation of the term "homotopic" Can someone explain me in descriptive words or maybe with an image, what homotopic actually means and what its relevance is?
Thank you for your time,
Chris
 A: Let $X$ and $Y$ be topological spaces, and let $F_0, F_1:X\rightarrow Y$ be continuous maps. We say that $F_0$ and $F_1$ are homotopic and denoted by $F_0 \simeq F_1$ if there exists a continuous map $H: X \times I \rightarrow Y$ (which is called a homotopy from $F_0$ to $F_1$) such that for all $x \in X$, 
$$H(x,0)=F_0(x),$$ 
$$H(x,1)=F_1(x).$$
If we think of the parameter $t$ as the time parameter which changes from 0 to 1, then the homotopy $H$ represents a continuous deformation of the map $F_0$ to the map $F_1.$
We know that a path in a topological space $X$ is a continuous map $f:I \rightarrow X$ such that $f(0)=x_0$ and $f(1)=x_1$. The points $x_0$ and $x_1$ are called initial point and terminal point of $f$, respectively. We say that $f$ is a path from $x_0$ to $x_1.$ For any two paths in the topological space $X$, we need a stronger relation between the paths in order to find the holes of the space $X.$
Let $f_0, f_1:I\rightarrow X$ be two paths in $X$. We say that $f_0$ and $f_1$ are path-homotopic and denoted by $F_0 \sim F_1$ if $f_0$ and $f_1$ have the same initial point $x_0$ and the same terminal point $x_1$ and there exists a continuous map $H: I \times I \rightarrow X$ (which is called a path-homotopy from $f_0$ to $f_1$) such that
$$ H(s,0)=f_0(s) $$
$$ H(s,1)=f_1(s) $$
$$ H(0,t)=x_0 $$
$$ H(1,t)=x_1 $$
for all $s \in I$ and for all $t \in I.$
In other words, the path-homotopy $H$ represents a continuous deformation of the path $f_0$ to the path $f_1$. Also end points remain fixed during the deformation. 
