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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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?
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1answer
99 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
136 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 & ...
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4answers
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Example of a finite non-commutative ring without a unity

Give an example of a finite, non-commutative ring, which does not have a unity. I can't think of any thing which fits this question. I was thinking $M_2(\mathbb{R})$ but it has the identity. Any ...
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0answers
56 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 ...
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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 ...
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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 ...
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2answers
825 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 ...
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0answers
24 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
18 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 ...
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0answers
45 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 $Y$? ($\sim$ ...
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1answer
26 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
60 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 ...
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3answers
333 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 ...
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0answers
56 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 ...
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5answers
721 views

Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
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1answer
44 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 ...
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9answers
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Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
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1answer
35 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 ...
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4answers
91 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
38 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 ?
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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 $\mathbb{T}$ ...
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1answer
111 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 ...
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1answer
44 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 ...
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2answers
387 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? ...
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1answer
104 views

The topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$.

I'm looking for an example of a topological space $X$ together with an equivalence relation $\sim$ where the product topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$ as a final ...
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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 ...
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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
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0answers
20 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 ...
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1answer
28 views

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere?

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere. I cannot think of one..
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1answer
25 views

Are surjective order-homomorphisms necessarily complete lattice homomorphisms?

Let $X$ and $Y$ denote complete lattice, and suppose $f : X \rightarrow Y$ is a surjective order-homomorphism. Does $f$ necessarily preserve arbitrary suprema, therefore being a complete lattice ...
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2answers
36 views

Does $(x-a)^n\in K[x]$ imply that $a\in K$?

Let $F$ be an extension field of $K$. Let $a\in F$ and $n$ be a positive integer. It is also given that the polynomial $(x-a)^n$ has all of its coefficients in $K$, i.e. $(x-a)^n\in K[x]$. Does it ...
2
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2answers
61 views

What are counter examples for these statements?

Question 1. Let $\{T_i\}_{i\in I}$ be a family topologies on a set $X$. Provide an example that $\bigcup T_i$ is not a topology on X. > Question 2. Let $X$ be a compact space ...
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0answers
38 views

Can an algebraic structure such that $x+a=x+b$, have solutions for all $a,b∈K$ exist?

Is there exist a algebraic structure $K$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b∈K$?
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0answers
56 views

Homeomorphisms of product spaces: an example [duplicate]

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to ...
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3answers
2k views

Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
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1answer
48 views

What is wrong with this “counterexample” of boundedness of weakly convergent sequences?

Weakly convergence sequences $\{u_n\}$ in a Hilbert space $H$ are bounded. Here is an attempted "counterexample". What is wrong with this? Let $H = \ell_2(\mathbb{N})$, and let $\{e_n\}$ be the ...
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1answer
21 views

Necessity of a hypothesis in Scheffé's lemma

Scheffé's lemma states that if $f_n$ is a sequence of Lebesgue integrable functions (i.e. $f_n \in L_1$) that converges almost everywhere to another integrable function $f$, then $\int |f_n - f| \, ...
2
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1answer
41 views

Product of two symmetric banded matrices - real eigenvalues?

Let $A$ and $B$ be real symmetric banded matrices but $AB$ is not symmetric. Are the eigenvalues of $AB$ real? A more specific case: let $D$ be a real diagonal matrix, $B$ real symmetric and banded, ...
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0answers
26 views

A locally connected metric space of first category

I wonder if there exists a locally connected metric space of first category. I proved a theorem which assumes a space to be connected locally connected hereditarily Baire and metrizable. I ...
2
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0answers
37 views

A real-valued separately continuous function discontinuous everywhere

I need an example of a real-valued separately continuous function that is discontinuous at each point. This is either a well-known folklore fact or a burning problem in separate versus joint ...
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0answers
19 views

F sigma subsets of the real line

I need an example of a nowhere dense $\mathcal F_\sigma$ subset of the real line that is not a countable union of perfects sets and not a countable union of pairwise disjoint closed sets.
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5answers
760 views

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere? I think it is probable because we can consider $$ y ...
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2answers
88 views

A function such that $f'(x)>0$, but not strictly monotone increasing.

This is Exercise 10.3.5 from Analysis Vol.1 by Terence Tao. Give an example of a subset $X \subset \mathbb{R}$ and a function $f: X \to \mathbb{R}$ which is differentiable on $X$, is such that ...
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1answer
64 views

Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
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1answer
322 views

Example of similar matrices $A$ and $B$ such that products $AB$ and $BA$ are not similar

I'm looking for the simplest possible example of square matrices $A$ and $B$ such that $A$ is similar to $B$, $AB$ is not similar to $BA$. Such an example should exist, but I would like to find ...
2
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2answers
57 views

Topological vector spaces vs. locally convex spaces

I'm taking a course on locally convex spaces and our lecturer mentioned that these form the most general collection of spaces on which one can still prove interesting theorems (like Hahn-Banach - ...
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1answer
39 views

Extending a compact operator to the entire Hilbert space

In a course I'm taking we defined compact operators as a linear mapping $H\rightarrow H$, where $H$ is a Hilbert space, that maps bounded sets to relative compact ones. The lecturer mentioned that the ...
6
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1answer
117 views

Example where Čech and derived functor cohomologies don't agree.

I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces. Is there a simple example of a topological space ...