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Jun
9
awarded  Supporter
Jun
9
answered $X$ and $Y$ are homotopy equivalent $\Leftrightarrow$ $\exists Z:$ $X,Y$ are strong deformation retracts of $Z$
May
7
awarded  Scholar
May
7
accepted A contractible mapping cone?
May
6
comment Showing that metric induces single unique topology on a finite set
Your argument shows that any metric defined on a finite set will induce the discrete topology on that set. Therefore, the discrete topology is the only one that can be induced by a metric on a finite set.
May
6
revised Topological Properties of a 3-Torus
improve the wording of a statement
May
6
revised What are some examples of a mathematical result being counterintuitive?
Example added to improve clarity
May
6
revised Topological Properties of a 3-Torus
Improve clarity
May
6
revised Topological Properties of a 3-Torus
Expanded answer
May
4
answered What are some examples of a mathematical result being counterintuitive?
May
4
awarded  Teacher
May
4
answered Topological Properties of a 3-Torus
May
3
comment A contractible mapping cone?
The motivation for this exercise is the following result: If two maps are homotopic, their mapping cylinders (and hence their mapping cones) are homotopy equivalent. Therefore, one can show that the dunce cap is contractible because it is easily seen to be homeomorphic to the mapping cone of $f$.
May
1
answered A contractible mapping cone?
May
1
revised A contractible mapping cone?
Clearing up the style, as I was able to provide an answer
Apr
30
comment A contractible mapping cone?
@Justin: Got it, thanks! I guess I need to take a complex analysis course. How did you construct $H$?
Apr
30
awarded  Editor
Apr
30
comment A contractible mapping cone?
@Justin: (Sorry, I don't seem to have enough reputation to comment on your answer or vote up your answer, which I would gladly do!) Thank you for answering. I understand that $f(e^{2\pi it})=e^{2\pi i(4t)} $for $0 \leq t\leq 1/2$. But why is that $f(e^{2\pi it})=e^{2\pi i(3-2t)}$ for $1/2\le t\leq 1$? If $z=e^{2\pi it}$, then $\bar{z}^{2}=e^{2\pi i(2-2t)}$, no? Therefore, it should be $f(e^{2\pi it})=e^{2\pi i(2-2t)}$ for $1/2\le t\leq 1$. Also, how did you construct the homotopy $H$? Some kind of reparametrization lemma?
Apr
30
revised A contractible mapping cone?
Expanding the details of the problem
Apr
30
suggested approved edit on A contractible mapping cone?