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I have been following Serge Lang's Intoduction to Complex Analysis at a Graduate Level and I met this theorem. I want to ask what does it mean for a function to be homotopic to a point? I am only familiar with paths being homotopic to each other but not to a point. Here is the extract

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If anybody could shine some light onto this I will really appreciate it.

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    $\begingroup$ "Point" here means "constant path at a point". $\endgroup$
    – Ben Steffan
    Jul 15, 2020 at 22:35
  • $\begingroup$ Hi @BenSteffan, by "constant path at a point" do you mean a path that consists of one point? $\endgroup$
    – Maths Wizzard
    Jul 15, 2020 at 22:38
  • $\begingroup$ Yes, MathsWizard, that's precisely what it means. This is true when the set $U$ is simply connected. $\endgroup$
    – Ted Shifrin
    Jul 15, 2020 at 22:40
  • $\begingroup$ Hi @TedShifrin, thank you for the comment. Could you clarify what you are referring to as being true for U simply connected? $\endgroup$
    – Maths Wizzard
    Jul 15, 2020 at 22:41
  • $\begingroup$ In a simply connected region, every closed path is homotopic to a constant path. Most students encounter this notion, at least intuitively, in multivariable calculus. $\endgroup$
    – Ted Shifrin
    Jul 15, 2020 at 22:43

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A path homotopy $f\to g$ is a homotopy $H=H(x,t)$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x).$ A path $f$ is homotopic to a point $c$ if it is path homotopic to the constant path $c(x)=c$ for each $x.$

Consider this example: let $f(x)=(\sin x,\cos x)$ be the path which traces out the unit circle in $\mathbb R^2$. Then the path homotopy $H(x,t)= tf(x)$ has $H(x,1)=f(x)$ and $H(x,0)=0$ for each $x.$ It follows that $f$ is homotopic to the point $0$ since it is homotopic to the constant path at $0.$

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