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The first qusetion is about the following proposition.

Let $E_1$ and $E_2$ be closed discs with boundaries $B_1$ and $B_2$, respectively. Then, any continuous map $\mathit f$ : $B_1$$\to$$B_2$ can be extended to a contiuous map F : $E_1$$\to$$E_2$. If $\mathit f$ is a homeomorphism, then so is F.

Here the closed disk is defined to the topological space which is homeomorphic to the unit closed disk(denoted by $E^2$) in $\Bbb R^2$.

The writer claims that it suffices to prove the case when $E_1=E_2=E^2$ and $B_1$, $B_2$ be the unit circle. But I don't know how to prove this.

The second question is about the following proposition.

Let $E_1$ be the closed disk. Let $E_2$ denote the quotient space of $E_1$ obtained by identifying a closed segment of the boundary of $E_1$ to a point. Then $E_2$ is again a closed disk.

The writer says that in view of the preceding proposition(in my first question), it suffices to prove this assertion for the case of particular closed disk and a particular segment on the bounday of that disk. Again I got stuck. I don't know how to use the preceding proposition.

I think that these two are quite simple questions, but I don't know how to solve them...Thanks for any help or hints!

GTM 73
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  • For the first question: for any disk, say $E_1$ with boundary $B_1$, there exists a homeomorphism from $E_1$ to $E^2$, the restriction of which to $B_1$ is a homeomorphism to the unit circle. Do you see why this is true? Do you see why it implies the writer's claim? – Greg Martin Jun 15 '19 at 07:50
  • Yes. That is true, which is an exercise on Massey's book. But I still don't know how to prove...The point is that, I don't know whether I should explicitly construct such a map F from E^2 to E^2 or prove the existence directly. If the claim works, I know what to do next. – GTM 73 Jun 15 '19 at 12:23
  • For the second question, you need to show there is a homeomorphism $S^1 \rightarrow S^1$ that takes the first segment to the second. You should be able to write out the formula. Then you use the prior result to extend this homeomorphism to a homeomorphism of the disk. Then it is a general fact that if you have a relative homeomorphism $(X,A) \cong (Y,B)$, then $X/A \cong Y/B$. – Connor Malin Jun 16 '19 at 17:17

1 Answers1

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Let $f:S^n \rightarrow S^m$ be a continuous function. Let $x \in D^{n+1}$ and denote its magnitude by $r$ and its unit direction vector by $v$. Define an extension of $f$ by $f(x)=rf(v)$. I leave it to you to check continuity, injectivity in the case the original function is injective, and surjectivity if the original functions is surjective.

These last two imply the result about homeomorphisms since $D^n$ is compact and $D^m$ is Hausdorff.

Adam Chalumeau
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Connor Malin
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  • dude i'll give u a green check mark if yah read this and fix it or tell me it's good https://math.stackexchange.com/questions/3274147/let-a-be-a-loop-in-mathbbrp2-which-is-non-trivial-in-h-1-mathbbrp2/3274204?noredirect=1#comment6734047_3274204 –  Jun 26 '19 at 13:59