Let $X$ be a set with the discrete topology and let $\pi_n(X)$, as usual, be the set of homotopy classes of maps $f : S^n \rightarrow X$ that map the base point $a$ to the base point $b$. I'm trying to prove that, in this case, $\pi_n(X)$ is singleton for all $n>0$, that is, every two base-point-preserving maps are equivalent. I think I have to show that, given $f$ and $g$ base-point-preserving maps, there is $F:S^n\times[0,1]\rightarrow X$ continuous such that $F(x,0)=f(x)$, $F(x,1)=g(x)$ and $F(a,t)=b$, $\forall \ x\in S^n$ and $\forall \ t\in[0,1]$. But I couldn't notice how to get that using the discrete topology in $X$ and I'd like to get a hint, for kindness. Thank you!
If $X$ is discrete then points are connected components (i.e. $X$ is totally disconnected). In particular if $Y$ is connected then every continuous map $Y\to X$ is constant. Since $S^n$ is connected then the thesis trivially follows since there is only one continuous function $S^n\to X$ that fixes base points.