Intereting Posts

Normal bundle of a section of a $\mathbb{P}^1$-bundle
Prove that $x-1$ divides $x^n-1$
infimum of a sequence
Free resources to start learning Discrete Mathematics
Is the fundamental group of every subset of $\mathbb{R}^2$ torsion free?
Prove that if $g^2=e$ for all g in G then G is Abelian.
Additive primes
stability of a linear system
Find $J: S^2\rightarrow \mathbb{R}$, if given $I:S^2\rightarrow \mathbb{R}$, s.t. $I(\vec{a})=\int_{S^2}\vec{n}\cdot\vec{a} J(\vec{n}) ds$.
Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?
Left and right ideals of $R=\left\{\bigl(\begin{smallmatrix}a&b\\0&c \end{smallmatrix}\bigr) : a\in\mathbb Z, \ b,c\in\mathbb Q\right\}$
How to check, whether the formula is a tautology
Does there exist a constructible (by unmarked straightedge and compass) angle that cannot be quintsected?
Equivalent limit definition
Intersection of open affines can be covered by open sets distinguished in *both*affines

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, $\beta_{n}\rightarrow b \in B$.

- Universal property of initial topology
- When is the image of a proper map closed?
- How can I prove that this function is continuous at $0$?
- Why do we need Hausdorff-ness in definition of topological manifold?
- Is this a metric on R?
- How many points does Stone-Čech compactification add?

- If the sequence satisfies the property lim$_{n\to \infty}(a_n-a_{n-2})=0$, prove that lim$_{n\to \infty}\frac{a_n-a_{n-1}}{n}=0$.
- Archimedean Property - The use of the property in basic real anaysis proofs
- Unconventional (but instructive) proofs of basic theorems of calculus
- A question about a proof of the “Least Upper Bound Property” in the Tao's Real Analysis notes
- Lie group, differential of multiplication map
- Why second countable for definition of manifold?
- Free cocompact action of discrete group gives a covering map
- Evaluate $\int\sin(\sin x)~dx$
- What is the meaning of the expression $\liminf f_n$?
- A sufficient condition for a function to be of class $C^2$ in the weak sense.

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.

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- Why are punctured neighborhoods in the definition of the limit of a function?
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