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The following lemma is called Borsuk lemma which can be found in Munkres' topology (Lemma 62.2).

(Borsuk lemma) Let $a$ and $b$ be points of $S^2$. Let $A$ be a compact space, and let $f:A\to S^2\setminus\{a,b\}$ be a continuous injective map. If $f$ is nulhomotopic, then $a$ and $b$ lie in the same component of $S^2\setminus f(A)$.

Now I need to show that if we does not assume $f$ to be injective, then the conclusion need not hold. I don't know how to find an example, the hypothesis that $f$ is nulhomotopic seems difficult to satifsy.

Xiang Yu
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  • Hint: If $A$ is a disjoint union of two contractible spaces, then any map from $A$ to $S^2$ is null homotopic. – stewbasic Jun 27 '16 at 04:26

1 Answers1

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Let $A = [0,1]$ (which is compact). Let $a$ and $b$ be any points NOT on the equator of $S^2 = \{(x,y,z)\in \mathbb{R}^3: x^2 + y^2 + z^2 = 1\}$.

Let $f:A\rightarrow S^2$ be defined by $f(t) = (\cos(2\pi t), \sin(2\pi t),0)$.

Then $f$ is not injective since $f(0) = f(1)$. Further, $f$ is nullhomotopic because $[0,1]$ is contractible.

Finally, $S^2\setminus f(A)$ has two components: The northern and southern hemisphere, and $a$ and $b$ can be in either component, independently.

Jason DeVito - on hiatus
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  • To conclude $f$ is nulhomotopic, I think we also need the fact that $S^2$ is path-connected. – Xiang Yu Jun 27 '16 at 04:37
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    I don't follow. By definition, $[0,1]$ is contractible means there is homotopy $F$ between the identity map of $[0,1]$ and a constant map. If $Y$ is any topological space at all and $f:[0,1]\rightarrow Y$ is continuous, then the homotopy $H(x,t) = f(F(x,t))$ is a composition of continuous things, so is continuous, and is a homotpy between $f$ and a constant map. – Jason DeVito - on hiatus Jun 27 '16 at 04:42
  • Yes, you are right. To show all maps are homotopic to the same constant map, we need the range space to be path-connected. – Xiang Yu Jun 27 '16 at 04:45
  • Yes, but that happens for free: Since $[0,1]$ is path connected, so is any continuous image of it. – Jason DeVito - on hiatus Jun 27 '16 at 04:47