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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Surjective étale morphism between normal schemes

If I have $X,Y$ noetherian schemes, locally of finite type over a field $k$, normal and connected, can I somehow conclude that a surjective étale morphism $X \to Y$ is finite? I have seen some counter ...
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Counterexaples for non-uniform sensitive dependence on initial conditions?

A map $f : X → X$ on a metric space is said to exhibit sensitive dependence at the point $x_0 ∈X$ if there exists a number $\epsilon$ > 0 with the following property: For every neighborhood $N$ of $...
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If $X$ is first-countable then a net converges when a subsequence converges?

Let be $X$ and we assume that $(x_\lambda)_{\lambda\in\Lambda}$ is a net such that there exists a cofinal and increasing map $\varphi$ form $\Bbb N$ to $\Lambda$ such that $\big(x_{\varphi(n)}\big)_{n\...
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-1 votes
1 answer
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If $x_n\ge 0$ for each $n\in\Bbb N$ and $\sum_{n\in\Bbb N}x_n=0$ then $x_n=0$ for each $n\in\Bbb N$ necessarly. [duplicate]

Let be $(x_n)_{n\in\Bbb N}$ a not negative sequence of real number and we suppose that $$ \sum_{n\in\Bbb N}x_n=0 $$ So if the last identity holds then it must be $$ x_n=0 $$ for any $n\in\Bbb N$ ...
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0 answers
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$\mathbb{R}_l^2$ is not normal. [duplicate]

Let $\mathbb R_l$ be the Sorgenfrey line, i.e. $\mathbb R$ with the lower limit topology. I want to prove that $\mathbb{R}_l\times \mathbb{R}_l$ is not normal. I do not want to use Urysohn's lemma ...
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1 vote
0 answers
49 views

Can circuit $f'$ replace circuit $f$?

Assume Peter and Paul develop a circuit. Paul tries to convince Peter that $f$ may be replaced by $f'$ to save hardware resources. $$\begin{aligned} f &:= \neg x_0 \land \neg x_1 \land \neg x_2 \...
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0 votes
0 answers
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Non-Abelian group without subgroup in this particular form. [duplicate]

So for my math class (Introduction to Groups and Rings) and I have encountered the following question: $\hspace{15px}\|$Find an example of an non-abelian group $G$ such that $H=\{g\in G : |g|<\...
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4 votes
1 answer
73 views

Necessity of the hypotheses of the alternating series test

The alternating series test states: Given a decreasing non-negative sequence $a_n$ such that $$\lim_{n\to \infty} a_n =0,$$ the series $$ \sum_{n=1}^\infty (-1)^na_n $$ converges. Does anyone know of ...
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0 votes
1 answer
27 views

Is the following characterisation of measurable functions true?

I am self studying measure theory.I measure theory it is often important to check if a function is measurable.If the function is continuous then it is measurable of course.But if the function is not ...
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3 votes
2 answers
112 views

Show that if $A$ and $B$ commute, then $B$ commutes with $e^A$

Show that if $A$ and $B$ commute, then $B$ commutes with $e^A$ For the first one I have $$e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} +\dots$$ I believe I can pull out $$I + A + \frac{A^2}{2!} + \...
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0 votes
1 answer
78 views

Is there example of equalizer that is not injective?

we know that in general case every arrow is not function also in the $Div \mathbb{A}b$ there is example of monic that is not injective. Is there example of equalizer that is not injective ?
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1 answer
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Quotients of real projective spaces

I am trying to determine whether or not $M := \mathbb{R}P^4/\mathbb{R}P^2$ is a manifold. I believe it is clear that $M$ is second-countable. However, I'm not not sure about Hausdorffness and being ...
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1 vote
0 answers
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Counterexample for Mecke equation in higher dimensions

I am currently reading the book Lectures on the Poisson Process by Gunter Last and Mathew Penrose. (The book can be found here.) I have a question about an exercise in the book's 4th chapter (Exercise ...
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0 votes
1 answer
38 views

Examples of counterintuitive regulated functions (solution verification)

Questions I am looking for some counterexamples for properties that of functions that are commonly mistaken for being true in cases where they do not hold in general. The specific cases I am thinking ...
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Example of a sequence of positive Riemann integrable functions with given properties

I would like to find a sequence $(f_n)_{n\in\Bbb N_0}$ of functions $f_n:[0,1]\to(0,+\infty)$ with the following five properties (if such a sequence exists): $f_n$ is Riemann integrable $\forall n\...
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1 vote
1 answer
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Prove or disprove that $x_0\in\operatorname{cl} Y$ iff there exists a sequence $x_n$ in $Y\subseteq X$ converging to $x_0$ when $X$ is sequential.

The following is an exercise from Elementos de Topología General by Ángel Tamariz Mascarúa and Fidel Casarrubias Segura. First a couple of definitions: Definiton A topological space $X$ is Frechét-...
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1 vote
0 answers
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Prove that $f:X\to Y$ is continuous iff $x_n\to x$ implies that $f(x_n)\to f(x)$ when $X$ is sequential

Definition A topological space $X$ is said sequential when a subet $Y$ is not closed if and only if there exists a sequence $(x_n)_{n\in\Bbb N}$ in $Y$ convergint to a point $x_0$ of $X\setminus Y$. ...
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1 vote
1 answer
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Prove or disprove that $\operatorname{cl}\big(\operatorname{cl} Y\setminus\{x_0\}\big)=\operatorname{cl}Y$ when $x_0$ is an accumulation point for $Y$

So given a set $Y$ of a topological space $X$ I ask to prove or to disprove if the identity $$ \operatorname{cl}\big(\operatorname{cl} Y\setminus\{x_0\}\big)=\operatorname{cl}Y $$ holds when $x_0$ is ...
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0 votes
0 answers
32 views

Ranks of Matrices

I was asked to prove/disprove the following claim: "Let $A$ and $B$ be two real square matrices, with the same order, $AB$ and $BA$ are not the zero matrix. Hence $\operatorname{rank}(AB) = \...
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1 vote
3 answers
38 views

If $K$ is an invertible matrix so $K+K^{-1}$ is diagonalizable, is $K$ necessarily diagonalizable?

Let $K$ be an invertible $n\times n$ matrix so $K+K^{-1}$ is diagonalizable. Is $K$ necessarily diagonalizable? Can't come up with a counter example, but also can't prove it... Here's my attempt so ...
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2 answers
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Is the boundary of singleton set in $T_1$ space is empty?

The singleton set in $T_1 $ topological space is closed so, I wonder how i use this information to prove that boundary of this set is empty.
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1 vote
1 answer
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Can you disprove this counterexample to the diagonal lemma?

I was looking at the Diagonal Lemma or Fix point theorem which states in every Theory $T$ every formula with one variable $ B(n) $ has a fix point: $T \vdash G \leftrightarrow B(\# G)$. Where $\#F$ ...
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3 votes
1 answer
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A sort of converse of Banach-Steinhaus theorem.

$(X, \|•\|) $ and $(Y, \|•\|') $ be two normed space. $\begin{align} {\scr{B}}{(X, Y) }&=\{T\in {\scr{L}}{(X,Y)}: T \text{ is bounded } \}\end{align}$ $\|T\|_{op}=\sup\{\|Tx\|':\|x\|\le 1 \}$ ...
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7 votes
6 answers
588 views

What are some good, elementary and maybe also interesting proofs by induction?

I am hosting a one-time class/talk on the concept of infinity for some (talented) high-school students. I want to teach them about proof by induction and I want them to do some exercises (you learn ...
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  • 691
6 votes
2 answers
70 views

Let $1=n_0<n_1<\ldots$ be an increasing sequence of positive integers. True/False: $\sum_{i=1}^\infty\frac{n_{i+1}-n_i}{n_{i+1}}$ diverges to $\infty$ [duplicate]

Let $1=n_0<n_1<n_2<\ldots$ be an increasing sequence of positive integers. Is it true that $\displaystyle\sum_{i=0}^{\infty} \frac{n_{i+1}-n_i}{n_{i+1}} $ diverges to $+\infty?$ For example, ...
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1 vote
1 answer
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Are the sets $\Bbb R^+\times\{0\}$ and $\mathcal G(f):=\{(x,y)\in\Bbb R^2:xy=1\wedge x\in\Bbb R^+\}$ homeomorphic?

So I would like to prove that the sets $\Bbb R^+\times\{0\}$ and $\mathcal G(f):=\{(x,y)\in\Bbb R^2:xy=1\wedge x\in\Bbb R^+\}$ are homemorphic and to do this I tried to use the following ...
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9 votes
2 answers
524 views

Example of an algebra that is a Banach space but not a Banach algebra

I'm looking for an example of a space $\mathbb{A} $ such, $\mathbb{A} $ is an algebra; $\mathbb{A}$ is equipped with a norm that makes it a Banach space; $\mathbb{A}$ is not a Banach algebra, i.e., ...
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1 vote
0 answers
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≞ (measured by symbol) example usage

I found this new equal sign that is named ' measured by ' (≞), and tried to find an exaple usage of it to no avail. I am a little uncertain about how to use it, and have two hypotheses. I think it is ...
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  • 111
1 vote
3 answers
89 views

If $E,F\subseteq X$ are homeomorphic set then is $E$ open/closed if $F$ is open/closed?

Let be $X$ a topological space and we suppose that $E$ and $F$ are homeomorphic though a map $f$ from $F$ to $E$. So I ask to me if $E$ is open/closed when $F$ is open/closed but unfortunatley I was ...
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1 vote
1 answer
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Prove that the function $φ:C(I)\rightarrow\Bbb R$ defined as $φ(f):=f(1)$ for any $f\in C(I)$ is not continous with respect the $L_2$ topology.

If $C(I)$ is the set of continous function on $I:=[0,1]$ then it is a well know result that the positions $d_\infty(f,g):=\sup\{|f(x)-g(x)|:x\in I\}$ $\langle f,g\rangle:=\int_0^1fg$ for any $f,g\in ...
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0 votes
0 answers
20 views

Neighborhood in K-topology

Consider $X = [0,1]$ with the Euclidean topology and $Y=[0,1]$ with the K-topology. By K-topology we mean the topology with the following basis $$ \left\lbrace (a,b); a<b \right\rbrace \cup \left\...
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2 votes
1 answer
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A simple example of a $G$-principal bundle that cannot be lifted to an $F$-principal bundle along an epimorphism $F\to G$.

I am trying to understand spin structures, in particular how they may fail to exist. To start with, I would like to see a most simple example of a $G$-principal bundle that cannot be lifted to an $F$-...
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1 vote
1 answer
51 views

Show $\forall h\in H,\forall g\in G, ghg^{-1}=h$ is not definition of normal subgroup.

Let $G$ be a group and $H$ be subgroup of $G$. $H$ is normal subgroup if only if $$\forall h\in H,\forall g\in G, ghg^{-1}\in H.\tag{1}$$ I want to know a lot of examples $H$ which does not satisfy $$\...
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0 votes
0 answers
37 views

Find specific example for where the case works (topic: isomorphism and normal subgroups)

Consider groups $G_i, K_i, F_i$ such that $K_i \triangleleft G_i$ and $G_i / K_i\cong F_i$ , for $i = 1, 2$. In each case, find an example with the given properties or prove that no such example ...
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  • 409
0 votes
3 answers
61 views

Behavior of power series on the circle of convergence

Is there a power series which is conditional convergent everywhere on the circle of convergence? I know that a power serier is absolutly convergent at one point of its circle of convergence if and ...
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1 vote
1 answer
37 views

Is there an inseparable subset of the complex plane?

Is there an inseparable subset of the complex plane ?I proved that all separable subsets form a sigma-algebra,and a rectangle is obviously separable, so a counter example would not a Borel set.
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1 vote
2 answers
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integrability of $f$ on $\Bbb R$ does not necessarily imply the convergence of $f(x)$ to $0$ as $x → ∞$ [duplicate]

I read that integrability of $f$ on $\Bbb R$ does not necessarily imply the convergence of $f(x)$ to $0$ as $x → ∞$. I am pretty sure that I also read somewhere something saying the opposite. Since I ...
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10 votes
2 answers
129 views

Can a space with $\pi_1(X)=F_2$ have a nontrivial covering space which is homeomorphic to $X$?

Any finite sheeted connected cover of the circle is again homeomorphic to a circle. On the group level, this is consistent with the fact that subgroups of $\mathbb Z$ are isomorphic to $\mathbb Z$. ...
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1 vote
0 answers
10 views

An example of a fundamental domain of a group of isometries with nonnull boundary?

I am reading some lecture notes on Fuschian groups, and one theorem given is that if $\Gamma$ is a group of isometries of the hyperbolic plane, and if $F_1$ and $F_2$ are fundamental domains of $\...
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  • 1,686
0 votes
1 answer
67 views

Is uniform convergence required for a continuous limit function?

Consider the sequence $(f_{n})_{n=1}^{\infty}$ of continuous functions on $I = [0, \infty)$ defined recursively by $f_{1}(x)=x, f_{n}(x)=x+\int_{0}^{x}f_{n-1}(t)\sin(x-t) dt, \forall n\geq 2$. This ...
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4 votes
2 answers
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Are all compact subsets of $\mathbb R^{\mathbb R}$ separable?

The title is the question. I should perhaps add that $\mathbb R$ has its usual topology and the product has the product topology, i.e., $\mathbb R^{\mathbb R}$ is the space of all functions from $\...
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  • 10.5k
1 vote
2 answers
165 views

$\int_0^1|f(x)|dx=0$ if and only if $f(x)=0$ for all $x\in[0,1]$ [duplicate]

If $f:[0,1]\rightarrow\Bbb R$ is a continuous function and not negative function such that $$ \int_0^1f(x)dx=0 $$ then could exist $x_0\in [0,1]$ such that $f(x_0)$ is not zero? I know this result but ...
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0 votes
0 answers
39 views

Is the sum of norms/metrics a norm/metric on the product space?

Let be $\mathfrak X:=\Big\{\big(X_i,\nu_i\big):i\in I\Big\}$ a FINITE collection of normed spaces. So I would like to know if the position $$ \nu(x):=\sum_{i\in I}\nu_i(x_i) $$ for all $x\in\prod_{i\...
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1 vote
1 answer
35 views

Finding an arbitrary pair of sets $U,V$ that is a separation of $A \in X$ and will satisfy $U \cap V \cap (X - A) \neq \emptyset$

Say I have some topology $\mathcal{T}$ on $X = \lbrace a,b,c \rbrace$ and a disconnected subset $A \subset X$. Can it be true that any arbitrary pair of sets - say $U$ and $V$ - that is a separation ...
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6 votes
2 answers
269 views

Uniform continuity and derivatives [closed]

Is there any example of a three times differentiable and uniformly continuous function $f: \Bbb R \rightarrow \Bbb R$ such that $f^{(3)}$ is bounded but $f^{(2)}$ isn't?
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0 votes
1 answer
95 views

Please explain these counter-interpretations to these Natural Deduction arguments

I have attached a screenshot of the arguments, but here they are written out just in case. I am able to prove when sentences are equivalent, but could somebody please explain the counter-...
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7 votes
1 answer
89 views

Examples of continua that are contractible but are not locally connected at any point

A continuum is a compact, connected, metrizable space. What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of ...
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0 votes
1 answer
27 views

Does every subalgebra of a finite dimensional commutative $k$-algebra has a complementary subalgebra?

Does every subalgebra $B$ of a finite dimensional commutative $k$-algebra $A$ has a complementary subalgebra (i.e. there exists a subalgebra $C$ such that $A=B\otimes C$)? The analogous statement for ...
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2 votes
0 answers
67 views

What are some examples of beautiful invariance?

I have been recently intrigued by the 2011 IMO Problem 2 which describes a windmill process exemplified in this video by 3Blue1Brown. The question is fundamentally about invariance in that the number ...
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0 votes
0 answers
29 views

If $X$ is subset of $\mathbb{R}$, '$X$ is closed set' is equivalent to '$X=Y'$ for some countable set $Y \subset \mathbb{R}$'? [duplicate]

If $X$ is subset of $\mathbb{R}$, '$X$ is closed set' is equivalent to '$X=Y'$ for some countable set $Y \subset \mathbb{R}$'? ($Y'$ means the limit point's set of $Y$) In many example, I can find ...
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