Intereting Posts

Construction of a regular pentagon
Prove the empty set is closed for a metric space (X,d).
If $|A| > \frac{|G|}{2} $ then $AA = G $
What is the primary source of Hilbert's famous “man in the street” statement?
Can we solve this using stars and bars?
Solving infinite sums with primes.
Cauchy sequence in a normed space
Is the boundary of a boundary a subset of the boundary?
System of $24$ variables
Solving simple congruences by hand
If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved
Functions determine geometry … Riemannian / metric geometry?
Hensel Lifting and solving with mods
Why is the empty set bounded?
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$

Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B=\{a. b: a\in A,b\in B\}$ closed?

If $p$ is an adherent point in $A\cdot B$, exists a sequence $p_{n}=\alpha_{n} \beta_{n}$, $\alpha_{n} \in A$ and $\beta_{n}\in B$, such that $p_{n}\rightarrow p$. If $B$ is compact, $\beta_{n}\rightarrow b \in B$.

- Cantor set is boundary of regular open set
- Is every manifold a metric space?
- An equivalent definition of open functions
- Gelfand-Naimark Theorem
- A question about the Zariski topology
- Topology - Quotient Space and Homeomorphism

- Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?
- About fractional iterations and improper integrals
- If $\sum{a_k}$ converges, then $\lim ka_k=0$.
- measurability with zero measure
- At what points is the following function continuous?
- Meaning of $f:\to\mathbb{R}$
- Subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?
- How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals?
- How to show $\lim_{n \to \infty} a_n = \frac{ + + + \dotsb + }{n^2} = x/2$?
- Can Path Connectedness be Defined without Using the Unit Interval?

Let $A=\mathbb{N}$ and $B=\{\frac{1}{n}:n \in \mathbb{N} \}\cup\{0 \}$ , then A is closed and B is compact but $A.B$ is no closed because $\sqrt{2}$ is an adherent point of $A.B$ , and $\sqrt{2} \notin A.B$ .

$C=A\times B$, catch hold a sequence $c_n=a_n\times b_n$ such that $c_n\to c$

we need to show $c=a\times b$ for some $a\in A, b\in B$

since $b_n\in B$ and $B$ is closed, bounded so it has a convergent subsequence say $b_{n_k}\to b$

now $a_n={c_n\over b_n}\in A, a_n\to {c\over b}$ but $A$ is closed so ${c\over b}\in A$, call it $a$

so $c=a\times b$ Done!

In harmonic analysis we have if $G$ is topological group and $A$ and $B$ be subset of $G$ , $A$ closed and $B$ is compact then $A.B$ is closed.(proof in principle of harmonic analysis written by anton Deitmar , Lemma 1.1.4). and compact condition is needed i.e. if A and B are closed then AB is not closed.

If we add the condition $0 \not \in B$, I think that the answer is yes.

- A convergence problem in Banach spaces related to ergodic theory
- Limit involving tetration
- Construct matrix given eigenvalues and eigenvectors
- Infinitely many primes of the form $4n+3$
- Proving $\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1$
- Are all the norms of $L^p$ space equivalent?
- Showing that $\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$
- Exponential Word Problem s(x)=5000−4000e^(−x)
- How to show that $\sum_{n=1}^{\infty} \frac{1}{(2n+1)(2n+2)(2n+3)}=\ln(2)-1/2$?
- Calculation of $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^2n}{3^n\left(m\cdot 3^n+n\cdot 3^m\right)}$
- Expected number of matches when two shuffled rows of $52$ playing cards are lined up
- Why are these estimates to the German tank problem different?
- Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?
- Set of Linear equation has no solution or unique solution or infinite solution?
- $15x\equiv 20 \pmod{88}$ Euclid's algorithm