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Questions tagged [examples-counterexamples]

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may disagree with the actual situation that follows from the definitions. This tag should be used only in conjunction with another tag to clearly specify the subject.

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Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$?

Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$? I was thinking we could approximate any set from inside by a closed set . This need not true from outside. So ...
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3answers
41 views

Constructing a weird probability measure

Is it possible to construct a probability space $(\Omega,\mathcal{A},\mathbb{P})$ such that $\mathcal{A}$ is uncountable, there are uncountable events with probability $>0$ and there are also ...
6
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1answer
84 views

Doubt in Understanding Lebesgue measure.

I am studying Measure theory form Stein and Shakarachi:Real Analysis. I come across observation regarding the outer measure. For any $E\in R^d$ $m_*(E)=\inf m_*(O)$ where $O $ is the open set ...
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1answer
30 views

What matrix has only negative or zero real part for all the eigenvalues?

Say $X \in \mathbb{R}^{m\times m}$, Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part? What I conjecture The following $X$ has only negative ...
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1answer
85 views

Intuition behind a counterexample to $|A+A|\leq |A-A|$, where $A$ is a finite set

Define $$A+A=\{a+b:a,b \in A\}, A-A = \{a-b:a,b \in A\}$$ Then prove or disprove the following $$|A+A|\leq |A-A|$$ Intuitively, it should be true, as $$a+b=b+a$$ $$a-b \neq b-a$$ ...
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1answer
36 views

Counterexample for Hausdorff Assumption is removed

I had the following fact: $A\subset X,f:A\to Y$ , where Y is Hausdorff space .f is continuous function on A. If there is continuous extension of f as $g:\bar A\to Y$ exist .Then this extension is ...
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4answers
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Provide a counterexample: If $n^2-1$ is divisible by $5$, then $n$ is divisible by $2$ or $3$

Provide a counterexample: If $n^2-1$ is divisible by $5$, then $n$ is divisible by $2$ or $3$ My book doesn't have an answer to this question, but I think it's $n=6$. Since $6^2-1=35$, which is ...
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2answers
37 views

Explicit example of a direct sum over two simple groups

I want to understand the notion of a direct sum properly. I learn best via examples so it would very helpful if I could get a concrete example using two simple groups. for examples suppose we want to ...
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1answer
13 views

Is every completely regular topology induced by some proximity?

A proximity space is a set endowed with a relation defining a notion of when two subsets are near or far apart. A proximity space induces a topology, and such a topology is always completely regular. ...
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1answer
30 views

Are metrics uniformly equivalent if and only if they have the same zero-distance sets?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are ...
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2answers
36 views

Is a completely regular space whose convergent sequences are eventually constant discrete?

If $X$ is a metrizable topological space where the only convergent sequences are eventually constant sequences, then $X$ must be a discrete space. But I'm interested in whether something stronger is ...
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1answer
22 views

Trace of a filter

Here are the definitions of the extension and trace of a filter (paraphrased from IM James' book "Topological and Uniform Structures"): Let $\mathcal{F}$ be a filter on the set $X$. For each set $X'...
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2answers
29 views

Star-convex is not convex?

Could anyone give a subset which is star-convex in $\mathbb{R^{2}} $ but is not convex?
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2answers
35 views

Is closedness assumption is essential in following argument?

$F$ is closed subset of $X$. $A$ is any subset of $X$. $X$ is a metric space. The topology on $X$ is induced by the metric. Define $A_n= \{ x\in F^c\cap A\mid d(x,F)\ge \frac{1}{n}\}$ Then $\bigcup_{...
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2answers
66 views

Measurable function that does not map a set of measure zero to a set of measure zero

I'm looking for an example a measurable function that does not map a set of measure zero to a set of measure zero. (we are discussing absolutely continuous functions in class, and the fact that they ...
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0answers
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Smooth function $f$ with large Hausdorff dimension of $\{x: \ x=0,\ \nabla f(x)\ne0\}$

Is it possible to construct a function $f:[0,1]^2 \to \mathbb R$ twice continuously differentiable (or in the Sobolev space $H^2((0,1)^2)$ such that the set $$ \{ x: \ f(x)=0 , \ \nabla f(x)\ne 0\} $$ ...
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1answer
31 views

Stable and unstable manifolds that are tangent to each other in a continuous dynamical system?

I am thinking of a scenario/ examples where the stable and unstable manifold of an equilibrium of a continuous dynamical system are tangent to each other? Any examples/ plots would be helpful?
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2answers
45 views

Non-existence of the potential function

I am wondering why this theorem is not true when $(f,g)$ are defined on a more general open set $U$ which is not necessarily the entire plane or some disc. What is an example of a vector field $$F(x,...
3
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1answer
27 views

Open set in terms of nets

Let $X$ be a topological space, and $U\subseteq X$ be a subset of $X$ with the following property: For every convergent net $x_\alpha\to x$ in $X$ such that $x\in U$, there exists an $\alpha$ such ...
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0answers
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Non-conformal metrics on vector bundles where $\nabla g=\omega(\cdot) g$

Let $E$ be a smooth vector bundle over a manifold $M$ ($\dim M \ge 2$), equipped with a metric $g$ and a connection $\nabla$, such that $\nabla_X g=\omega (X) g$ for every vector field $X$ on $M$. ($\...
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1answer
31 views

Find a function for which integral does not exist but converges as a limit of a sequence

Find an example of a function $f:[0,\infty)\rightarrow \mathbb{R}$ integrable on all intervals such that $\lim_{n\rightarrow \infty}\int_0^n f$ converges as a limit of a sequence, but such that $\...
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0answers
317 views
+200

Conjectured primality tests for specific classes of $k\cdot b^n \pm 1$

Can you provide proofs or counterexamples for the claims given below? Inspired by Lucas-Lehmer-Riesel primality test I have formulated the following two claims: First claim Let $P_m(x)=2^{-m}\...
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0answers
167 views

Conjecture: $n>2$ is prime iff $\sum_{k=1}^{n-1}\left(3^k-2\right)^{n-1} \;\equiv\; n \cdot 2^{n-1}-1 \pmod{\frac{3^n-1}{2}}$

This question is closely related to: Conjectured primality test Can you provide a proof or a counterexample for the following claim : Conjecture. Let $n$ be a natural number greater than $2$. Then ...
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2answers
32 views

A function that is continous but non constant between two particular topological spaces [closed]

Find a non-constant function between $X,\tau_1$ and $(X,\tau)$ and $(X,\tau')$ where $\tau=\{X,12,34,\emptyset\}$ and $\tau'={X,123,12,1,\emptyset}$. $f:(X,\tau)\to (X,\tau')$ I know that I need ...
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1answer
64 views

given a continuous function from $f:\mathbb{Q}\to \mathbb{Q}$ [duplicate]

Given a continuous function from $f:\mathbb{Q}\to \mathbb{Q}$ there exists a continuous function $g:\mathbb{R}\to \mathbb{R}$ such that g restricts to f on $\mathbb{Q}$ I could think of one example ...
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0answers
140 views

Congruence satisfied by primes and only by primes II

This question is closely related to: Congruence satisfied by primes and only by primes Can you provide a proof or a counterexample for the following claim : Let n be a natural number greater than ...
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1answer
23 views

Example of $\mu_n$ absolutely continuous to Lebesgue but $\mu$ is not

Give an example of a sequence of probability measures $(\mu_n)_{n\in\mathbb{N}}$ on $\mathbb{R}$ equipped with Borel $\sigma$-algebra converging in weak topology to a probability measure $\mu$ such ...
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1answer
17 views

Converging almost everywhere in finite space but not in p-th mean

Give example of a finite measure space and a sequence of functions which converges $\mu$-a.e but not in p-mean, for any $p\geqslant 1$. I was thinking of the following spaces $([0,1],\mathscr{B_{[0,1]...
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3answers
72 views

Problem when convert $\sqrt{A}+\sqrt{B}=\sqrt{C}+\sqrt{D}$ to A+B=C+D.

This is what my lecturer taught me. If you have $\sqrt{A}+\sqrt{B}=\sqrt{C}+\sqrt{D}$ You can easily convert to $A+B=C+D$ or $AB=CD$ And then he gave me an example. $\sqrt{8x+1}+\sqrt{3x-5}=\sqrt{...
5
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1answer
60 views

Smallest subgroup generated by a subset of a group.

If$\ G$ is a group, and the set $S=\{a,b\}$ is a subset of $\ G$, can we say that the smallest subgroup of $\ G$ generated by $\langle a,b\rangle$ will always be either $\langle a \rangle$, $\langle b\...
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4answers
2k views

“Numbers” bigger than every natural number

In the book Understanding analysis, by Abbot, when discussing the Archimedean property, the author states that there are ordered field extensions of $\mathbb{Q}$ that include "numbers" bigger than ...
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1answer
25 views

Duality Theorem. An example where the dual is feasible but the primal is not

Can anyone provide an example where dual is feasible but the primal is not? This is a linear programming question.
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3answers
99 views

Can the union of uncountable infinite sets be a countable infinite set?

Can the union of uncountable infinite sets be a countable infinite set? If there is such a set, I would be grateful if you answer the question by giving an example. If the question is too simple, I ...
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1answer
57 views

Find continuous functions $f,g$ such that $g\circ f $ is closed and continuous but neither $g$ nor $f$ is closed map.

Find continuous functions $f,g$ such that $g\circ f $ is closed and continuous but neither $g$ nor $f$ is closed map. Find continuous functions $f,g$ such that $g\circ f $ is open and continuous ...
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2answers
69 views

What are some of the interesting examples of functions that can not be described symbolically? [closed]

Here is one I that I like. I got it from "Introduction to Topology and Modern Analysis by G. F. Simmons" Consider the function of the real variable $x$ defined as follows: for each real number $x$ ...
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1answer
33 views

Easy example of a herbrand structure

Can someone give me an easy example of a Herbrand structure? I can't really visualise the difference between a Herbrand and a normal structure.
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2answers
48 views

Does $L_1$ convergence of continuous functions imply pointwise convergence?

Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $x\in[0,1]$? I'm pretty sure the answer is no, ...
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1answer
17 views

Example of open and bounded subset of $\Bbb R$ that is not totally bounded

An interval $(a,b)$ is totally bounded in $\Bbb R$. Would $\bigcup\limits _{n\ge1}(1/2^n,1/{2^{n-1}})$ or $\bigcup\limits _{n\ge1}(1/\alpha^n,1/{\alpha^{n-1}})$ for $\alpha\in\Bbb R_{>2}$ be good ...
2
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1answer
37 views

Definition of a structure

I found the following definition for a structure in my math course: A structure $X$ consists of a non-empty set $D_x$, the universum of $X$ and the attribution of values $r^x$ to non logical-symbols $...
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0answers
22 views

Lp convergence for all p doesn't imply a.s. convergence counter-example check

I was trying to find a counter-example of a sequence of random variables that converge to 0 in $L_p$ for all $p$, but not almost surely. I was using $X_1, X_2, \dots$ independent and $$\mathbb P(X_n =...
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1answer
9 views

Non-constant function with 0 Lipschitz semi-norm

Suppose we have a bounded metric space $(X,d)$. We say a function $f:X\to \mathbb{R}$ is Lipschitz if $|f|=\sup_{\substack{x\neq y\\x,y\in X}}\frac{\left|f(x)-f(y)\right|}{d(x,y)}<\infty$. This ...
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3answers
90 views

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $ G\cong N \times G/N. $ [duplicate]

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $$ G\cong N \times G/N. $$ I tried to prove this claim, but then it seems that since $G$ is abelian then every ...
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3answers
47 views

How to construct examples of maps that are open or closed or continuous but not the others? [duplicate]

I came across this exercise, the main problem for me are the restrictions, i need to find examples for maps $f: \mathbb {R}^2 \rightarrow \mathbb {R}^2$ or subsets of them such that $f$ is only one or ...
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2answers
48 views

Show by means of an example that $\lim_{x \to a} [f(x)g(x)]$ may exist even though neither $\lim_{x \to a} f(x)$ nor $\lim_{x \to a} g(x)$ exists. [closed]

This is particularly a hard problem to solve, "a" has to be a defined number. We can't really pick any two functions simply like $\frac{1}{x}$ and any other to exemplify. Please help me with it
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1answer
57 views

Does this limit exist? $\lim_{n\to\infty}\frac{φ(f(n))}{ψ(g(n))}$

EDİTED: Is this statement true? Suppose $\left\{f,g,\varphi, \psi \right\}:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$ and for $n\to\infty$ $\left\{f(n),g(n),\varphi(n), \psi(n) \right\}\...
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1answer
28 views

Logical mistake in a proof of Dirichlet's test

While writing a proof of Dirichlet's Test, [ in which, a required condition is that $\displaystyle{\int_a^B\phi(x)dx}$ is bounded $\forall B>a]$, I made a logical mistake, which I noticed upon ...
2
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1answer
70 views

Is there any relationship between growth rate and amenability?

Let $G$ be a finitely generated group, I'm interested in whether there is any relationship between amenability of $G$ (as a discrete group) and its growth rate. To make the question more precise let's ...
2
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0answers
49 views

An example of a Banach algebra satisfying given conditions

Is there a non-commutative non-unital Banach algebra $A$ for which $aa_0 -a_{0}a$ lies in the annihilator of $A$ for any $a\in A$? Here $a_0$ is an element of $A$ not belonging to its centre $Z(A)$. ...
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2answers
35 views

Give an example of a space which is separable but no proper subspace of it is separable. [closed]

I need an example of a topological space which is itself separable but no proper subspace of it separable.
3
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1answer
58 views

A connected, but not path-connected, space whose fundamental group depends on the basepoint

It is a well known result that the fundamental group of a path-connected space is independent (up to isomorphism) of the choice of the basepoint. Can someone provide an explicit example of a connected,...