Tagged Questions

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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0answers
95 views

Please give me an example of the algorithm where $\Theta$ will be equal to $e^n$

Please give me an example of the algorithm where $\Theta$ or $O$ will be equal exactly to $e^n$ . The algorithm should not be simple counting from 0 till $e^n$ . It should be a clear relation of two ...
11
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1answer
131 views

If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?

Let $X$ be a path connected space and $Y$ be a topological space. Let $f:X\rightarrow Y$ be a function such that for every path $\tau:\mathbb{I}\rightarrow X$ , $f\tau:\mathbb{I}\rightarrow Y$ is ...
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1answer
49 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.
10
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1answer
134 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?
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0answers
22 views

A Lindelof non-scattered space $X$ which is not an extention of $\mathbb R$

Is anyone familier with an example for a Lindelof non-scattered topological space $X$ which is not an extention of $\mathbb R$ (with Euclidean topology). I am looking for an example which is not a ...
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1answer
79 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 $f(x,y)=\sin(xy^2)$. ...
4
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3answers
74 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 ...
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2answers
55 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
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1answer
69 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 ...
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1answer
63 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?
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2answers
57 views

Groups reluctant to have infinite subgroup

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
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1answer
61 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. ...
2
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2answers
111 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
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1answer
55 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 $F$ ...
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2answers
59 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.
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1answer
54 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!
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1answer
43 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 ...
2
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1answer
51 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.
1
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1answer
82 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 ...
6
votes
1answer
126 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
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0answers
84 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 $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?
2
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1answer
87 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 ...
2
votes
1answer
120 views

Equicontinuity of a pointwise convergent sequence of monotone functions with continuous limit

I was looking at this question, and trying to come up with a counterexample. After thinking about it, I thought the following might be true: Claim: let $\{f_n\}$ be a sequence of continuous, ...
5
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1answer
87 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
1answer
103 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 ...
2
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0answers
34 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 ...
4
votes
2answers
76 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 ...
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vote
3answers
140 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 & ...
1
vote
1answer
120 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...
4
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0answers
59 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 ...
5
votes
0answers
80 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 ...
1
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1answer
40 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 ...
1
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0answers
53 views

mid-point convex but not a.e. equal to a convex function

I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpniski's theorem from which we can deduce that for ...
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0answers
19 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 ...
7
votes
1answer
56 views

Is a topological space determined by its components and their quotient?

Given connected topological spaces $X_i$ and a totally disconnected space $Y$, is there a unique topological space $X$ with components homeomorphic to $X_i$ and $X/\sim$ homeomorphic to $Y$? ($\sim$ ...
2
votes
1answer
28 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 $G$?
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2answers
71 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
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0answers
68 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 ...
3
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1answer
54 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
41 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
92 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. ...
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1answer
28 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
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1answer
43 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
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1answer
60 views

Counterexample: Convergence in finite dimensional distributions does not imply weak convergence

I'm working at the following exercise: Give an example of a sequence of stochastic processes $(\mathbb{X}^n)_{n\geq 1}$ such that the finite dimensional distributions converge to the finite ...
1
vote
1answer
169 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 ...
6
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2answers
401 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
25 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
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1answer
44 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. $\limsup_{x\to \infty} f(x)=\infty$
2
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1answer
61 views

If $|\operatorname{Aut}_KF|=3$, must we have cube roots of unity?

Let $K$ be a field of zero characteristic. Let $F$ be a finite dimensional extension field of $K$ such that $|\operatorname{Aut}_K F|=3$. Must the equation $x^2+x+1=0$ have a root in $F$ ? Thank you
0
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0answers
22 views

Example of a Problem Made Easier with Skew Coordinates

I'm looking for an example of a problem which would be hard (or significantly harder) to solve in orthogonal coordinate systems, or at least the Cartesian coordinate system, but is reduced to an ...