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!
The set $\pi_n(X)$ of homotopy classes is singleton if $X$ is given the discrete topology
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general-topology
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homotopy-theory
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4If $X$ is discrete, then all continuous maps from connected spaces to $X$ have something in common that makes the exercise rather easy. – 2017-01-12
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1No, if $X$ is discrete, every subset is clopen, and so every continuous map from a connected space to $X$ is constant. So for $n > 0$ there is only one continuous map on $S^n$ that maps the base point to the base point. – 2017-01-13
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0Thanks for your help and sorry for my unfamiliarity with the subject. That is a rather easy exercise, like you said. – 2017-01-13
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0When you see the crucial point it's easy. But seeing the crucial point is always a nontrivial part. – 2017-01-13
1 Answers
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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.