I'm trying to solve a problem in Lee's topology book. In this book, a closed n-cell $D$ is any space homeomorphic to the closed unit ball. Then $\mathrm{Int} D$ , resp. $\partial D$ are the images of the open unit ball, resp. the boundary of the closed unit ball under such a homeomorphism. Given a continuous function $f: \partial D \rightarrow \left [ 0,1 \right ]$, he wants one to show that $f$ extends to a continuous function $F: D \rightarrow \left [ 0,1 \right ]$ that is strictly positive in $\mathrm{Int} D$. Obviously, it suffices to show this for a closed unit ball and I was thinking something like $$\left \|x  \right \|f\left(\frac{x}{\left \| x \right \|}\right) + \dfrac{1 -\left \| x \right \|}{2},$$ but that does not quite do it. Tietze's extension theorem is not in that book (and in fact is a weaker result). Any ideas? Thanks in advance.
I'm working in Lee's book on topological manifolds and have gotten stumped on the first question in chapter 5, the chapter on cell complexes.  The problem is:   Let $D$ and $D'$ be two closed cells not necessarily of the same dimension.     Show that any continuous map $f:\partial D \to \partial D'$ can be extended to a continuous map $F:D \to D'$ such that $F(Int \; D) \subseteq Int \; D'$. Given points $p \in Int \; D$ and $p' \in Int \; D'$, show that $F$ can be chosen so $F(p) = p'$. Show that if $f$ is a homeomorphism then $F$ can be chosen to be a homeomorphism.   I proved the first part roughly as follows: First suppose $D$ and $D'$ are convex and each contain $0$ in their interior (in their respective ambient spaces).  Then every element other than $0$ in $D$ can be expressed uniquely in the form $\lambda q$ where $q \in \partial D$ and $\lambda\in (0,1]$ (an equivalence relation on the chords connecting $0$ to boundary points partitions $D- \{0\}$).  I then define the map $F(\lambda q) = \lambda f(q)$ which is continuous since if $\lambda_n q_n \to \lambda q$ in $D$ then $F(\lambda_n q_n) \to F(\lambda q)$.  Lastly $\lambda q \in Int \; D$ implies $\lambda &lt;1$ and so $\lambda f(q)$ is an interior point of $D'$ since $f(q)$ is a boundary point and $D'$ is convex.    Now if we suppose $D$ and $D'$ are arbitrary closed cells there are homeomorphisms $g_1: \overline{\mathbb{B}^n} \to D$ and $g_2: \overline{\mathbb{B}^m} \to D'$ (where possibly $m=n$) then we have that $g_2^{-1}\circ f \circ g_1$ is a continuous map between the boundaries of two closed balls and so by the first part can be extended to a continuous map $F:\overline{\mathbb{B}^n} \to \overline{\mathbb{B}^m}$.  The mapping $g_2 \circ F \circ g_1^{-1}$ is a continuous map that satisfies the desired claim.    My issue is that parts 2 and 3 don't seem (to me at least) to follow that easily from the proof I constructed.  Maybe my proof is wrong and I'm not seeing it, but more likely I think that I'm missing the spirit in which Lee is intending for us to approach this problem.
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Thanks, Chan I want to buy a home, but I'm not interested in using a realtor as a buyer's agent.  I know that in MLS systems, when a realtor lists a house they specify what commission a buyer's agent will be paid.  Since I am the one who is bringing a qualified buyer to the listing agent, I feel that I would should get the commission (even if it is in the form of a discount on the house).    Essentially, the seller would get the same amount of money at closing... the listing agent would get the same commission as they normally would for showing the house to someone with a buyer's agent... but I would pay less for the house.  Is there a tactful way to negotiate this with the listing realtor?  For example, should I make a side contract with the realtor to pay me the commission at closing?  Or perhaps I can specify in my offer that they should reduce their fees to the seller by the commission amount if the offer is accepted, in order to make my offer more attractive?  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