I am trying to solve a problem, as mentioned in the title:
Given two continuous functions $f,g : X \to S^n $ such that $|f(x) - g(x)|<2$, $\forall x $, then $f$ is homotopic to $g$, i.e. $ f \sim g $.
What I am trying to solve this by using that for two continuous functions $ f,g : X \to S^n $ such that $f(x) \neq -g(x),\, \forall x $, then $ f \sim g $. And for that, I need to conclude $f(x) \neq -g(x)$ from $|f(x) - g(x)|<2$. But I have not succeeded yet....
If anyone has some other approach or solution, please provide... Thanks
If $S^n$ denotes the unit sphere then notice that $|v - (-v)| = |2 v| = 2$ for every $v\in S^n$. Therefore if the condition "$\forall x\in X\ |f(x) -g(x)| <2 $" holds then in particular $\forall x\in X\ g(x) \neq -f(x)$.