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!
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.