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 ...

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6
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1answer
58 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
5
votes
2answers
124 views

Give an example of a function $f$ satisfying $\lim_{x\to 0}(f(x)f(2x))=0$,but $\lim_{x\to 0}f(x)$ does not exists

Question: Give an example of a function $f$ satisfying the condition $$\lim_{x\to 0}(f(x)f(2x))=0$$ and such that $$\lim_{x\to 0}f(x)$$ does not exists. I think this question have many example. But ...
13
votes
2answers
688 views

Do differentiable functions preserve measure zero sets? Measurable sets?

Consider Lebesgue measure on $\mathbb{R}$ and let $f:\mathbb{R}\to\mathbb{R}$ be differentiable. Does $f$ necessarily preserve measure zero sets? Does $f$ necessarily preserve measurable sets? ...
3
votes
1answer
50 views

What are some examples of non-commutative monoids that are both idempotent and self-distributive (on both sides)?

In the presence of the axioms for a commutative monoid, idempotency is equivalent to self-distributivity. Proof. Suppose a commutative monoid is idempotent. Then: $$x(yz) = xxyz = (xy)(xz)$$ On the ...
10
votes
1answer
149 views

Are elements commutative when subgroups are?

Suppose G is a group and for every $H_1,H_2\le G$, $H_1H_2=H_2H_1$. Is $G$ abelian?
0
votes
1answer
72 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.
4
votes
3answers
88 views

Non-isomorphic algebraic structures such that each surjects homomorphically onto the other

Off the top of my head, I cannot think of any algebraic structures $X$ and $Y$ such that each surjects homomorphically onto the other, yet $X$ and $Y$ are non-isomorphic. What are some examples of ...
0
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1answer
149 views

An example in Spivak's Calculus on Manifolds (chain rule).

Spivak gives an example which has step that is giving me some problems to get it, even if it's supposed to be trivial. Spivak says: Let $f:\mathbb{R}^2\to\mathbb{R}$ given by $f(x,y)=\sin(xy^2)$. ...
2
votes
2answers
65 views

Convergence of integrals but $\int_a^b|f_n(x)-f(x)|dx$ does not converge to $0$

Can I have an example of a sequence of continuous functions $(f_n)_n$ and a continuous function $f:[a,b]\to \mathbb{R}$ such that $$\int_a^bf_n(x)dx\to\int_a^bf(x)dx,$$ when $n\to+\infty$, but ...
2
votes
1answer
73 views

An operator such that $\|A\|^2 \neq \|A^2\|$

The question asks for a bounded linear operator on a Hilbert space satisfying the condition in the title. This is what I came up with: Let $A_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a 90-degree ...
-1
votes
1answer
83 views

Why set is not equal its closure minus its boundary? [closed]

Why $ \Omega \neq \bar{\Omega} \setminus \partial \Omega $ ? Can somebody show any counterexample?
7
votes
1answer
305 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
3
votes
2answers
76 views

Groups reluctant to have infinite subgroup

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
1
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1answer
413 views

A counter-example to Dini's Theorem (after removing a hypothesis)

Recall Dini's Theorem: Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. ...
7
votes
1answer
109 views

Definition of the normalizer of a subgroup

Let $G$ be a group and $H$ a subgroup of $G$. Is there any counterexample to the assertion $N_G(H):=\{g\in G\mid gHg^{-1}=H\}=\{g\in G\mid gHg^{-1}\subset H\}$? Thanks!
2
votes
2answers
173 views

Conjecture: if $a+b$ and $ab$ are rational, $a$ and $b$ are rational

I can't find a rigorous proof but I have a feeling it's true. Informal argument: Suppose $a+b$ and $ab$ are rational, $a$ and $b$ are irrational (since just one can't be irrational). Then $a$ and ...
2
votes
1answer
78 views

Fundamental theorem of calculus necessary assumption

The fundamental theorem of calculus is stated as follows: Let $f$ and $F$ be real-valued functions defined on a closed interval $[a, b]$ such that the derivative of $F$ is $f$. That is, $f$ and $F$ ...
1
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2answers
89 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
2
votes
1answer
58 views

Can any “relevant” topological spaces be decomposed into an uncountable product?

Can any "relevant", as meaning generally useful topological spaces be decomposed into an uncountable product of other topological spaces with the product topology? Many thanks in advance.
3
votes
1answer
114 views

Example of a function f that is Generalized Riemann Integrable, but its square is NOT Generalized Riemann Integrable.

I am reading a section about Generalized Riemann Integral (Kurzweil-Henstock), and there was a problem on that section to provide an example of a function $f$ on $[0,1]$ that is Generalized Riemann ...
1
vote
1answer
146 views

An example of when a product in a category may not exist.

My question is about the category of finitely generated Abelian groups; in particular, I want to show, by definition, that there exists a set of objects $T$ in this category for which there is no ...
1
vote
1answer
118 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed (or just compact) 4-manifolds. Are there any closed (or compact) ...
4
votes
0answers
92 views

A sequence of subsets of an infinite group.

Is there an infinite group $G$ such there is not any sequence $(A_n)$ of its subsets such that always $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?
4
votes
1answer
258 views

Unbounded variation but differentiable everywhere

A function with bounded variation is differentiable almost everywhere. There are also functions with unbounded variation that are differentiable almost everywhere (e.g. take ...
4
votes
1answer
268 views

Some thinking about the Dini' s Theorem.

In the Dini's Theorem, On the compact set $K$,if $f_n$ is a sequence of monotone increasing or decreasing continuous functions, i,e $f_n(x)\leq(\geq) f_{n+1}(x)$ for all $n$ and all $x$, converges ...
5
votes
1answer
102 views

What makes “the topos $\mathbf{M}_2$” such a good counterexample?

I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with ...
4
votes
0answers
164 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
3
votes
2answers
98 views

Decomposing an element into product of elements of finite order

If $G$ is a group and $g, h\in G$, it is possible that $g$ and $h$ have finite order, yet $gh$ has infinite order. For example, in Algebra: Chapter 0 by Paolo Aluffi, Exercise 1.12, the following is ...
0
votes
4answers
7k views

Why is the empty set a subset of every set? [duplicate]

Take for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?
2
votes
1answer
394 views

Topological counterexample: compact, Hausdorff, separable space which is not first-countable

I need an example for a compact, Hausdorff, separable space which is not first-countable. I tried to look for it for some time without success...
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3answers
184 views

A counter-example to differential function but not twice differential

Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & ...
4
votes
0answers
108 views

Does there exist an infinite solvable group with no normal abelian subgroups?

This is impossible if $G$ is finite and solvable, because then $G$ has a (nontrivial) minimal normal subgroup $A$, which can be shown (using a trick) to be abelian. I'm trying to mimic the same ...
1
vote
1answer
44 views

A group with bounded element orders and its minimal and maximal subgroups.

Let $n>1$ be an integer. Is there an abelian group $G$ with all elements of order less than $n$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains ...
5
votes
0answers
135 views

Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
18
votes
2answers
1k views

“Pseudo-Cauchy” sequences: are they also Cauchy?

I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample. Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
0
votes
0answers
28 views

maximal and minimal subgroups of torsion abelian groups

Is there a torsion abelian group $G$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains a minimal (non-trivial) subgroup of $G$. 2) every proper ...
2
votes
1answer
44 views

Minimal normal subgroups in a non-torsion group

Is there a group $G$ with an element with infinite order such that every non-trivial $N \unlhd G$ contains a minimal (non-trivial) normal subgroup of $G$?
1
vote
2answers
143 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
4
votes
3answers
626 views

Does a monotone function on an arbitrary subset of $\mathbb R$ always have at most countable number of discontinuity?

I know a monotone function of a closed and bounded interval can have at most countably many point of discontinuity. And hence a monotone function on $\mathbb R$ can have at most countably many point ...
4
votes
0answers
137 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
4
votes
1answer
113 views

Does paracompact Hausdorff imply perfectly normal?

That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially ...
2
votes
1answer
85 views

Counterexample for Palais-Smale condition

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when ...
3
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4answers
110 views

is $\mathbb{R}^2\setminus \{(0,0)\}$ homeomorphic to $S^1$?

I found an exam question asking to prove that a homeomorphism exists but I am quite doubtful that this is true. Can anyone verify this? I can easily prove that the quotient space is homeomorphic. ...
1
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1answer
141 views

Quotient of a locally compact space

I am looking for an example of a quotient of a locally compact space that isn't locally compact. Is there a not too complicated example ?
1
vote
1answer
29 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...
1
vote
1answer
489 views

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ always convergent? [closed]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ and $\sum \sqrt{a_na_{n+1}}$ always convergent? Either prove it or give a counter example. I am thinking ...
3
votes
1answer
51 views

$\mathcal{C}$ a family of connected sets with connected union. For all $C \in \mathcal{C}$ there is $C' \in \mathcal{C}$ with $C\cup C'$ connected.

It is easy to show that if the union of a finite family $\mathcal{C}$ of (more than one) connected sets is connected, then for any $C \in \mathcal{C}$ there must always be some other $C' \in ...
6
votes
2answers
554 views

Difference between being faithful and being injective on arrows

I'm studying the concept of faithful functors, but I cannot grasp the difference between being faithtful and being injective on arrows. Could someone explain the difference and provide some examples? ...
0
votes
1answer
35 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
0
votes
1answer
47 views

WHat would be an example of this function?

WHat would be an example of a Lebesgue integrable function $f:\mathbb{R}\rightarrow [0,\infty)$ satisfying: $f$ is continuous. $\limsup_{x\to \infty} f(x)=\infty$