Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Which of his books? Intro to topological manifolds? – Rudy the Reindeer Jan 17 '13 at 20:40
Why does the map not do it? – Stefan Hamcke Jan 17 '13 at 22:08
I agree with Stefan H., your proposed $F$ works. – Matemáticos Chibchas Jan 17 '13 at 22:22
Dear @user58664, I apologize for the rubbish answer I just deleted. It was late when I wrote it and it adds absolutely nothing to your already fine answer. – Rudy the Reindeer Jan 18 '13 at 19:36

F is not defined at the origin. By defining F(0) = 1/2, F becomes a solution to the problem.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.