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Jul
21
answered diagonalization of a matrix over finite fields
Jul
21
answered Proof that this set is not compact
Jul
21
answered Eigenspace, Diagonalizable, Direct Sum
Jul
21
revised Eigenspace, Diagonalizable, Direct Sum
Formatting
Jul
18
revised alternating series test convergence proof with Cauchy criterion
Formatting, grammar
Jul
17
comment homeomorphism non-example
I only argued that $\varphi$ is not a homeomorphism, not that other homeomorphisms aren't possible. This is also true - a topological space $X$ is called sequentially compact if every sequence $(x_n)$ of points in $X$ has a convergent subsequence. This is a topological notion and so preserved by homeomorphisms - if a space is sequentially compact then any space homeomorphic to it is also sequentially compact. The circle $S^1$ is sequentially compact while both $[0,1)$ and $\mathbb{R}$ aren't. Finally, like you wrote, locally, $\mathbb{R}$ and $S^1$ are homeomorphic.
Jul
17
reviewed Approve suggested edit on sum of the series $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$
Jul
17
reviewed Reject suggested edit on $f:X\rightarrow S^1$ a continuous map. $X$ a path-connected topological space.
Jul
17
reviewed Approve suggested edit on Characterize graph by its connectivity matrix
Jul
17
revised homeomorphism non-example
added 368 characters in body
Jul
17
comment The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$
I mean, yeah, we really need a cover such that $U = \cup_i B_{\frac{r_i}{2}}(x_i) = \cup_i B_{r_i}(x_i)$ for the argument to work so I modified my answer accordingly.
Jul
17
answered homeomorphism non-example
Jul
17
revised The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$
Corrected an error
Jul
17
comment The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$
Hmm, it seems that it doesn't work. If I take $U = \cup_{i} B_{\frac{r_i}{2}}(x_i)$, the function doesn't necessarily vanish on $E$. One needs to start a priori with a function that is positive on $B_1(0)$ and vanishes outside in order for the argument to work. I'm modifying it right now...
Jul
17
comment The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$
Yeah, I was a bit sloppy. I need $U = \cup_{i} B_{\frac{r_i}{2}}(x_i)$. This is always possible.
Jul
17
answered The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$
Jul
17
answered Differential on a product space as sum of differentials
Jul
14
answered $\lim_{n \rightarrow \infty} f_n(x) = n^2 \left( 1- \cos \frac{x^3 - 1}{n} \right)$
Jul
14
comment Eigenvalues of $A:\;A^2 +2A=0$
You're welcome.
Jul
14
comment Eigenvalues of $A:\;A^2 +2A=0$
Indeed. The Cayley-Hamilton theorem says that if you plug $A$ into the characteristic polynomial, you get $0$ (the zero matrix). If you have some polynomial $p$ such that $p(A) = 0$, it doesn't mean that $p$ is the characteristic polynomial.