# 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.

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### Counter example for Baire's Theorem

Theorem: Let $(X,d)$ be a complete metric space, and let $D_n, n\in \mathbb N$ be open, dense subsets of $X$. Then also $\bigcap_{n\in\mathbb N} D_n$ is dense in $X$. This statement is false if $X$ ...
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### Uniformly Integrability and Non-Tightness

I want to costruct a measure space $(X,\mathcal{F},\mu)$ and a $\mathcal{C}\subset\mathrm{m}\mathcal{F}$, where $\mathrm{m}\mathcal{F}$ be the set of extended real-valued measurable functions on $X$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### $\{f_n \}$ 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 ...
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### 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?
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### 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 ...
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### 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 $\kappa$...
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### 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 ? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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 $a\in\Omega$...