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Aug
7
comment General Isomorphism, for all algebraic structures
The "homomorphisms" in the category of topological spaces are the continuous functions, and they do not behave like homomorphisms in groups, rings,... See here.
Aug
7
comment General Isomorphism, for all algebraic structures
Take $f$ from $[0,2\pi)$ to the unit circle in $\Bbb R^2$ defined by $f(t)=(\cos(t),\sin(t))$.
Jul
20
comment Bounded Matrix-Vector Multiplication
$Ax$ is a vector, so its norm $\| Ax\|$ is a well-defined real number. If you are asking about the boundedness of the set $B=\{\| Ax\| :x\in \Bbb R^p\}$, then $B$ is bounded if and only if $A=0$.
Jul
16
comment Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?
@Siminore This is Theorem 30.7 of Willard's General Topology.
Jul
16
answered Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?
Jun
11
comment Prove that if $b \mid c$ then $ab \mid c$?
$5\cdot 5=4$ ${}{}{}$
Apr
27
awarded  Informed
Jan
26
comment Geometric proof of complex number equation
$e^{i\theta}$ lies in the unit circle, $|z-w|$ is the distance between $z$ and $w$, and $\sqrt{2}$ can be seen as the length of the diagonal of a square of side 1.
Jan
16
awarded  Yearling
Jan
14
comment Extremely hard and stimulating (undergraduate) real analysis $problems$
Problems in Real Analysis: Advanced Calculus on the Real Axis, maybe?
Jan
3
comment Deceptively simple divisibility problem
@tmrlvi but there is no $k$ such that $9=2a+b\mid (a+b)^k=7^k$.
Jan
2
comment How can I prove these generalizations of Weierstrass?
Glad it helped.
Jan
2
comment How can I prove these generalizations of Weierstrass?
Okay. $\lim_{x\to +\infty}f=+\infty$ iff for every real number $M$ there is a real number $b$ such that $f(x)>M$ if $x>N$ (this is the definition). Now, taking $M=f(a)$, we find a real number $b$ ($>a$ because the domain of $f$ is $[a,+\infty)$) such that $f(x)>f(a)$ if $x>b$. By the extreme value theorem (Weierstrass' th.), there is a real number $c\in [a,b]$ such that $f(c)\le f(x)$ for all $x\in [a,b]$. In particular, $f(c)\le f(a)$ and $f(a)<f(x)$ for $x\in (b,+\infty)$, so $f(c)\le f(x)$ for all $x\in [a,+\infty)$. Hence $c$ is a minimum of $f$.
Jan
2
comment How can I prove these generalizations of Weierstrass?
Which part do you find problematic to formalize?
Jan
2
comment How can I prove these generalizations of Weierstrass?
You can reduce the statements to the ones you know. For example, in $(1)$, by definition of $\lim_{x\to +\infty} f=+\infty$, there is some real number $b$ such that $f(x)>f(a)$ if $x>b$. Now you can find the minimum of $f$ in $[a,b]$.
Dec
20
comment elementary topology exercises reference
See Elementary Topology: Problem textbook
Dec
20
awarded  Constituent
Dec
16
answered Irreducible in $\Bbb Q[x]$
Dec
15
awarded  Custodian
Dec
15
reviewed No Action Needed Are there number systems corresponding to higher cardinalities than the real numbers?