The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [examples-counterexamples]

To be used for questions whose central topic is a request for examples where a mathematical property holds, or counterexamples where it does not hold. This tag should be used in conjunction with another tag to clearly specify the subject.

3,693 questions
Filter by
Sorted by
Tagged with
38 views

Find a group epimorphism $\mathbb{Z}_m \rightarrow \mathbb{Z}_{(m,n)}$ with kernel $n\mathbb{Z}_m$

Let $m$ and $n$ be positive integers and $\otimes=\otimes_\mathbb{Z}$. Denote the GCD of $m$ and $n$ by $(m, n)$. I proved $\mathbb{Z}_m \otimes \mathbb{Z}_n \cong \mathbb{Z}_{(m,n)}$ by considering ...
80 views

Center of a group with order $p^aq^b$

Are there any examples of groups, $G$, such that $|G|=p^aq^b$ (where $p$ and $q$ are distinct primes and $a,b\geq 1$) and $|\mathrm{Z}(G)|=p^a$? ($\mathrm{Z}(G)$ denotes the center of $G$) I ...
115 views

54 views

Construct $X$ s.t. it has a dense subset but elements in the complement of the subset have no sequence in the subset converging to them

Construct a topological space $X$ with the following property: There is a proper subset $Y$ of $X$ such that the closure of $Y$ is $X$ but if $c \in X-Y$ then there is no sequence in $Y$ converging to ...
101 views

Undecidable cardinality

Let $S$ be an infinite subset of $\mathbb{R}$ with the property that the existence of $S$ can be proved within ZFC (and in particular the definition of $S$ does not invoke the negation of the ...
50 views

Example of an experiment in which A, B, C are independent, but not pairwise independent

Can somebody give an example of process in which we have at least three events A, B, C and: P(A ∩ B ∩ C) = P(A) * P(B) * P(C) But A, B, C are not pairwise independent
71 views

Example of non regular surface

I was reading definition of surface in differential geometry book which defined as follows A subset $S\subset \mathbb R^3$ is regular surface if $\forall p\in S$ there is open set in S such that ...
52 views

$\{f_n \}$ is equicontinuous and pointwise converge to f is f uniformly continuous?

Let $\{f_n \}$ is equicontinuous and pointwise converge to $f$ now is $f$ uniformly continuous ? I can prove f is continuous but is this uniformly continuous? and we know every $f_n$ is uniformly ...
33 views

74 views

Easy counterexamples for each containment $Field\subsetneq ED \subsetneq PID \subsetneq UFD \subsetneq GCD \subsetneq Integral\space Domain$

$\mathbb Z$ could be used for an example of an Euclidean Domain that is not a Field. Can you give other easy examples for the other proper inclusions?
33 views

Does every linear injective open map between Banach spaces map closed sets to closed sets?

Let $X$ and $Y$ be Banach spaces and $T \in L(X,Y)$, i.e. $T$ is a linear continuous map from $X$ to $Y$. Further let $T$ be injective and open. Does $T$ map closed sets to closed sets? I know this ...
25 views

26 views

Concrete example of Lie algebra $\mathfrak{g}$ with $x$ s.t. $x$ is in the radical of $\mathfrak{g}$ but not the radical of the Killing form?

Let $\mathfrak{g}$ be a Lie algebra over algebraically closed field $k$ of characteristic $0$. The radical $R(\mathfrak{g})$ is the largest solvable ideal of $\mathfrak{g}$. The Killing form $\kappa$...
58 views

Is there is group which can not be made into ring with any operation? [duplicate]

I wanted to have example of group which can not be ring? I think if we have non abelian group with some operation we can not proceed to ring . Is it correct or it required to more argument ? ...
67 views

Is the interior of a Jordan curve a Borel set?

A Jordan curve is a continuous closed curve in the plane with no self-intersections. My question is, is the interior of a Jordan curve always a Borel set? If not, is the interior of a convex Jordan ...
62 views

Differentiability class: Example of maps that are $C^k$ but not $C^{k+1}$.

Is there a classical example for the fact that the differentiability class satify $$C^{k+1} \subsetneq C^{k}$$ I'm interested in the $C^{k+1} \neq C^{k}$, then is I'm looking for a classical ...
36 views

71 views

What is a good example of prefix induction?

the Wikipedia article for Mathematical induction introduces a few variations of the classic principle, such as the strong induction. The strong induction comes with a few examples, namely the closed ...
67 views

Counterexample in topological vector space.

Let's $X$ be a topological vector space. So it has two operations $+$ and $\times$. As we know these operations could be continous. But can it be that one of this operation is continous, but the ...
34 views

Basic theorem for solvable groups not true for nilpotent groups - counterexample.

it's my first question on MathStackExchange so please be tolerant. Let H be a normal subgroup of group G. If H and G/H are both solvable, then G is solvable. But H nilpotent and G/H nilpotent doesn't ...
59 views

Example of noncompact space in which every real valued continuous function on it is uniformly continuous

I wanted to find Example of non-compact metric space $(X,d)$ such that every real-valued continuous function is uniformly continuous My attempt: $X$ is an infinite set $d$ is a discrete metric. ...
66 views

Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown ...
52 views

Cantor's Intersection Theorem iff complete metric spaces?

If a metric space is complete, we know that the Cantor's Intersection Theorem holds. Does the converse also hold? And if not, what is a suitable counterexample for the same? Also, if the converse ...
39 views

Pointwise convergence vs convergence in measure [closed]

I would be very grateful if I could be given an example of a sequence convergent pointwise but it is not convergent in measure and an example of a sequence convergent in measure but it is not ...
285 views

“Counterexample” for the Inverse function theorem

In my class we stated the theorem as follows: Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ a $\mathscr{C}^1(\Omega)$ function. If $|J_f(a)|\ne0$ for some $a\in\Omega$...
38 views

Example of $\sum a_n$ convergent but $\sum \sqrt{a_{2^k}}$ divergent?

The problem is this: Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n\geq 0.$ I'm using the Cauchy's method: $\sum \frac{\sqrt{a_n}}{n}$ ...
55 views

A hyperbolic but nonstrictly hyperbolic linear conservation law

I need an example of hyperbolic but nonstrictly hyperbolic linear conservation law, but I am not getting one example in any books that I am studying. My main book is LeVeque (1) and the only example ...
65 views

Product of two sequential spaces is not sequential

The product of two sequential topological spaces is not necessarily sequential. Could you give an example confirming this? As each first-countable space is sequential and the operation of taking a ...
307 views

Example in Measure theory about Borel Measures

I need to show that there are distinct $\sigma$-finite measures in $B(\mathbb{R})$ that are the same in the open sets. Im not really seeing a good example cause every open set is a countable union of ...
39 views

Example of Linear functional on subspace which cannot have extension

I wanted to find example of linear functional on subspace of vector space which cannot be extended to whole vector space. I know that if we are in normed linear space then extension is always ...
51 views

Can the real numbers be embedded into all non-Archimedean real closed fields?

Every Archimedean real closed field is isomorphic to a subfield of $\mathbb{R}$. But I’m wondering if something in the opposite direction is true. Suppose that $F$ is a non-Archimedean real closed ...