which of the following statements are true?
1.every homeomorphism of the $2$-sphere to itself has a fixed point.
2.the intervals [$0,1$] and ($0,1$) are homeomorphic.
3.there exists a continuous surjective function from $S^1$ onto $\mathbb{R}$.
4.there exists a continuous surjective function from complex plane onto the non-zero reals.

my effort:

1.true as $2$-sphere is a compact set.
3.true as the function $f(α)=re^{|iα|}$ exist.
4. no idea.

can anyone help me please.....

  • $\begingroup$ For $2$ through $4$, the following two facts should come in handy at least once: (A) The continuous image of a compact set is compact. (B) The continuous image of a connected set is connected. $\endgroup$ Dec 10 '12 at 1:42
  • $\begingroup$ For number 2, what happens if you delete a point from $(0,1)$? What about $[0,1]$? $\endgroup$ Dec 10 '12 at 2:11
  1. $X$ compact does not imply that every homeomorphism of $X$ with itself has a fixed point. Consider $S^1$ where the homeomorphism is rotation by $\pi/2$. You need a better argument here.

  2. This is incorrect. Hint: think about counting special types of points in $[0,1]$ and $(0,1)$.

  3. Also incorrect. Think about compactness here.

  4. What is the "complane"? If the question is "Is there a surjective continuous mapping from the complex plane to the non-zero reals?", then you should draw pictures of both of these sets and look at them- what property does one have which the other does not? Can you use this?

  • $\begingroup$ can u explain question 4 please. $\endgroup$
    – poton
    Dec 10 '12 at 3:19
  • $\begingroup$ One is connected other don't $\endgroup$
    – Cloud JR K
    Dec 5 '18 at 18:25
  1. False. The antipodal map $ f: \mathbb{S}^{2} \rightarrow \mathbb{S}^{2} $ defined by $ f(x) = -x $ does not have a fixed point.

  2. False. The interval $ [0,1] $ is compact, but the interval $ (0,1) $ is not. Note Compactness is a topological invariant, i.e., it is preserved exactly by homeomorphisms.

  3. False. The unit circle $ \mathbb{S}^{1} $ is compact, but $ \mathbb{R} $ is not compact. Note The image of a compact space under a continuous map is also a compact space.

  4. False. The complex plane $ \mathbb{C} $ is connected, but the set $ \mathbb{R} \setminus \{ 0 \} $ of non-zero real numbers is not connected. Note The image of a connected space under a continuous map is also a connected space.


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