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Jan
25
awarded  Self-Learner
Jan
25
awarded  Notable Question
Jan
24
awarded  Nice Answer
Jan
18
answered Does free bimodule exist?
Dec
16
awarded  Popular Question
Dec
8
awarded  differential-geometry
Dec
6
awarded  Enlightened
Dec
6
awarded  Nice Answer
Nov
16
comment Prove or Disprove: if $\lim\limits_{n \to \infty} (a_{2n}-a_n)=0,$ then $\lim\limits_{n \to \infty} a_n=0.$
What about the constant sequence $a_n=1$?
Nov
8
comment First-order definition of “$f$ is continuous at $x$” using just $<$
@avid19 I see, is that the problem, that there are no such symbols in the language? But how is $\Bbb R$ to be understood in this context?
Nov
8
comment First-order definition of “$f$ is continuous at $x$” using just $<$
Wouldn't something like $$\varphi(x)=\forall\epsilon >0~\exists\delta>0~\forall y~ (x-\delta<y<x+\delta\rightarrow f(x)-\epsilon<f(y)<f(x)+\epsilon)$$ do the trick?
Sep
27
comment Is the homotopy category cartesian closed?
If you consider unbased paths, then there is a homotopy equivalence $X\simeq X^I$.
Sep
27
comment Is the homotopy category cartesian closed?
@MikeMiller ok, but why then does the OP say that the path space is isomorphic to the discrete set of path components? This doesn't seem to gel with a basepoint approach.
Sep
27
comment Is the homotopy category cartesian closed?
"the path space of a path-connected space is contractible" I very much doubt that's true. Am I wrong?
Sep
12
answered show that $\lambda(x+y+z)^2+2c(x^2+y^2+z^2)>0$ for any $x,y,z$ if and only if $c>0$ and $\lambda+\frac{2c}{3}>0$?
Sep
12
asked Existence of a measurable $\theta$ such that $\frac{f(y)-f(x)}{y-x}=f'(\theta_{x,y})$?
Aug
21
awarded  general-topology
Aug
12
awarded  Enlightened
Aug
12
awarded  Nice Answer
Aug
7
comment Describe a group $G$ that acts on a set $X$ of 4 elements such that the action of $G$ has 2 orbits.
An explicit example comes from geometry: $\mathrm{GL}_2(\Bbb F_2)$ acts on $\Bbb F_2^2$, a four element set, and has two orbits. Similarly, yet less interestingly, the multiplicative group of units $\Bbb F_4^\times\simeq \Bbb Z/3\Bbb Z$ acts on the four element set $\Bbb F_4$ by homothecies and has two orbits.