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comment Exterior measure of a subset $A \subset \mathbb R_n$ equals the measure of a$G_{\delta}$
By definition that $m^*(A)=\inf\{\,m(U)\,\vert\,U\supseteq A\,\}$
Aug
27
comment Solutions to Topology by Munkres
@fixedp But as far as I've experienced, some of them are quite nontrivial and need some ingenious thoughtful ideas.
Aug
27
comment 6.17 Theorem : Show that $f \ \ \in \mathfrak R(\alpha)$ if and only if $ f\alpha' \ \ \in \mathfrak R$ ( walter rudin)
It follows from Rudin's argument that the upper and lower (Darboux) integrals of $fd\alpha$ and $f\alpha'dx$ are the same.
Aug
27
comment Differential identity and wedge products
$f\omega$ is just $f\wedge\omega$ where $f$ is considered as a $0$-form, and $d(\omega_1\wedge\omega_2)=d\omega_1\wedge\omega_2+(-1)^p\omega_1\wedge d\omega_2$ where $\omega_1$ is a $p$-form.
Aug
27
comment $f\in C(\mathbb{R})$. What does it mean?
But it's usually denoted as $C^n(\mathbb R)$ or $\mathcal C^n(\mathbb R)$ as well as $\mathcal C^0(\mathbb R)$ without parentheses.
Aug
27
comment A question on short exact sequences.
I found learning diagram chasing isn't that easy for a beginner from books, but there're two materials which seems more accessible: Hatcher's Algebraic Topology, pp115, subsection Relative Homology Group, and Eisenbud's Commmutative Algebra, pp637, subsection A3.7.
Aug
27
comment Exterior measure of a subset $A \subset \mathbb R_n$ equals the measure of a$G_{\delta}$
Hint: Choose $U_n\supseteq A$ such that $m(U_n)\le m^*(A)+1/n$, and take $H=\bigcap_n U_n$. As usual, the outer measure is denoted as $m^*$.
Aug
27
answered Prove $a^3+b^3+c^3\ge a^2+b^2+c^2$ if $ab+bc+ca\le 3abc$
Aug
27
revised How to graph functions without a calculator?
deleted 1 character in body
Aug
26
comment Differentiability of non-analytic complex functions
Well, and by Looman-Menchoff theorem, a continuous complex function is holomorphic in some region if and only if it satisfies Cauchy-Riemann equations everywhere.
Aug
25
accepted Homology of a finite graph follows from Mayer-Vietoris sequence?
Aug
24
comment Homology of a finite graph follows from Mayer-Vietoris sequence?
@LeeMosher Thanks. I've edited to include these.
Aug
24
revised Homology of a finite graph follows from Mayer-Vietoris sequence?
added 104 characters in body
Aug
24
asked Homology of a finite graph follows from Mayer-Vietoris sequence?
Aug
24
revised A generalization of Jordan curve theorem to connected open sets in the plane
edited body
Aug
22
reviewed Approve suggested edit on Equality of subgroups $H \subseteq K \subseteq G$ with the same finite index in $G$
Aug
22
asked Cycles non-homologous but with the same winding number at each point outside?
Aug
21
accepted A generalization of Jordan curve theorem to connected open sets in the plane
Aug
21
accepted Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?
Aug
20
asked A generalization of Jordan curve theorem to connected open sets in the plane