Understanding Serge Lang's Definition of Homotopy

Question

I have been following Serge Lang's Complex Analysis text book and today I came across a chapter on homotopy. I have trouble visualising and honestly, understanding the definition that he has given in his book. Here is the definition from his book

Could somebody explain to me how I can visually interpret this? I would also be really grateful if someone had a graphic or visual that would illustrate what is meant in this definition. Any help will be appreciated.

Answer 1

By definition, $\psi(t,c)=\gamma(t)$. Since $\psi$ is continuous, if $c_1$ is slightly bigger than $c$, then $t\mapsto\psi(t,c_1)$ is a path which is close to $\gamma$. And if $c_2$ is slightly bigger than $c_1$, then $t\mapsto\psi(t,c_2)$ is a path which is close to the previous one. And so on, until you reach $d$. So, $\psi$ deforms $\gamma$ into $\eta$.

Answer 2

You can see each $f_x(t)=\psi(t,x)$ as a continuous family of paths, indexed by the points in the interval $[c,d]$. The conditions mean that the initial path $f_c(t)$ is precisely $\gamma(t)$ and the final path is $\eta(t)$.

Answer 3

Essentially, for every $s\in[c,d]$, $\psi_s:[a,b]\to U,~\psi_s(t)=\psi(t,s)$ is a path with the property that $\psi_c=\gamma$ and $\psi_d=\eta$. So you could also view it as a continuous* function $[c,d]\to P(U)$, where I call $P(U)$ the set of all paths in $U$. Viewing it this way, it might be clearer what is meant by "continuous deformation": We take an interval $[c,d]$ and assign a path to each element of the interval in a way that if $s,s'\in[c,d]$ are close, then the paths $\psi_s$ and $\psi_{s'}$ are close, and the starting path is $\gamma$, while the end path is $\eta$.

Here is a desmos example of a homotopy between a semicircle and a line, where you can play with a slider determining the parameter $s$. For $s=0$ you get the line as a path, and for $s=1$ you get the semicircle. $0<s<1$ gives semiellipses interpolating between the line and the circle.

* If we equip the set of paths with the right topology, but that's a technicality you don't need for the intuition behind the definition.