Ivan Shishkin, Rye (1878)

Problems/TopologyExerciseUnreviewed

Un exemple simple d’homotopie

by Ancient Tree·translated by Catalpa·
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Les applications f,g:R→Rf,g:\R \rightarrow \R données par f(x)=x2f(x)=x^{2} et g(x)=(x+1)2g(x)=(x+1)^{2} sont-elles homotopesEN?

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Le graphe de f (vert) et g (rouge).

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Solution by Ancient Tree · translated by Catalpa

Discussions0 useful votes

Oui. Ici, l’homotopie la plus naturelle entre ff et gg est simplement l’application H(t,x)=(x+t)2H(t,x)=(x+t)^{2}.

Elle est continue et vérifie H(0,x)=x2=f(x)H(0,x)=x^{2}=f(x), tandis que H(1,x)=(x+1)2=g(x)H(1,x)=(x+1)^{2}=g(x).

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