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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Moduli space of stable principal $G$-bundles on a compact Riemann surface.

Let $C$ be a compact Riemann surface. I'm looking for some references in order to try to understand what is the moduli space of stable principal $G$-bundles on $C$, where $G$ is a simple Lie group. ...
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2answers
88 views

Which groups $G$ has the property that for all subgroups $H$ , there is a surjective map from $G$ to $H$?

I tried many examples , but i can't find any counterexample . But I guess there are many counter examples , and specific sorts of groups or subgroups have this property (e.g abelian groups or normal ...
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4answers
251 views

Linear map between the same dimensional spaces

Let $V,W$ be a vector spaces of the same dimension $m$ and $f\colon V\to W$ be a linear map. I know that for finite $m$, $f$ is injective $\Leftrightarrow$ $f$ is surjective $\Leftrightarrow$ $f$ is ...
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2answers
384 views

Can compact sets completey determine a topology?

Suppose that $\tau_1$ and $\tau_2$ are two topologies on a set $X$ with the property that $K\subset X$ is compact with respect to $\tau_1$ if and only if $K$ is compact with respect to $\tau_2$. Then ...
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2answers
48 views

$\nabla ^2 G$ meaning

If $G$ is a function with three components. $G$ is actually a Green's function in my case, like $G( \textbf{x} , \xi)$ with $\textbf{x} = (x,y)$ and $\xi = (\xi _x, \xi _y)$ so I am guessing it is ...
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4answers
44 views

How can I provide a counterexample for this predicate logic problem?

I'm honesty still unsure of what a counterexample even is, and what I've found on isn't helping me much in the way of understanding. I'm hoping to get pointed in a correct direction. Predicates ...
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2answers
42 views

A function not differentiable at a point but whose derivative has a limit

Does there exist a function $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is continuous on a neighborhood of $0$, $f$ is differentiable at all $x$ close to $0$ except at $0$ itself, and $\lim_{x\to 0} ...
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0answers
21 views

Second derivative of a set in non-Fréchet space

In Fréchet (T1) topological spaces it's easy to prove that $A''\subseteq A'$, but the proof doesn't work without this assumption. What are some illuminating counterexamples when the space is not ...
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1answer
74 views

Counter example to Stone Weierstrass Theorem

If we miss some conditions of Stone Weierstrass Theorem, will this theorem still hold? I have come up with counter examples when we do not have compact metric space. But what if the function algebra ...
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0answers
34 views

Examples of Algebraic quantum groups

I am reading articles about algebraic quantum groups, which are defined (see A. Van Dael) as a regular multiplier Hopf algebra $(A,\Delta)$ for which there exists a non-zero functional $\varphi$ on ...
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4answers
3k views

$3\times 3$ matrix with no real eigenvalues

I was asked this question on my hw along with any $2\times2$ matrix with no real eigenvalue and any $4\times4$ matrix with no real eigenvalue. I got the $2\times2$ which is $$ \begin{bmatrix} 1 & ...
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1answer
59 views

Direct sum of two closed subspaces of Banach space is not closed

I'm looking for an example of two closed subspace of a Banach space (or even a Hilbert space) whose sum is not closed. We have $l^2$ as Banach space and $A$ and $B$ are closed subspaces of $l^2$ : ...
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0answers
50 views

Topological manifold but not “smooth”

I would like to know if there are examples of topological manifolds that not admit a smooth atlas, or more generally a differentiable structure of class $C^k$ for $k\geq1$. In the book "Introduction ...
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1answer
43 views

Can we have a continuous choice in the mean value theorem

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable. Must there exist a continuous function $g:\{(a,b)\in \mathbb{R}^2: a<b\}\rightarrow \mathbb{R}$ such that: For every two distinct real ...
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3answers
245 views

Counterexample: Continuous, but not uniformly continuous functions do not preserve Cauchy Sequences

I want to prove this: There exists a continuous function $f:\mathbb{Q}\to\mathbb{Q}$, but not uniformly continuous, and a Cauchy sequence $\{x_n\}_{n\in\mathbb{N}}$ of rational numbers such that ...
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1answer
40 views

Finding an example of an extension of length 2

I am working on extensions in general but for sake of simplicity we can assume it's a module here. I am interested in an extension of the form $$0\to B \to E_2\to E_1 \to A \to 0$$ which is an ...
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3answers
65 views

How to construct a sequence that has a subsequence convergent to every $k\in \Bbb{N}$?

How to construct a sequence $\{a_{n}\}^{\infty}_{n=1}$, such that for every $k\in \Bbb{N}$, $\{a_{n}\}^{\infty}_{n=1}$ has a subsequence convergent to $k$? A subsequence is such as $2,4,6,...$ in ...
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0answers
25 views

Example of a non semisimple Lie Algebra over $\mathbb{C} $ such that $ L' = L$ [duplicate]

I wanted to show that if $L$ is a finite dimension semisimple Lie Algebra over $\mathbb{C}$, then $L' = L$ ($L' = [L,L]$) but the converse is not true. I could prove the statement but am unable ...
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1answer
24 views

Equidecomposable examples

Decomposable: A set $S \subset \mathbb{R}^n$ is decomposable in $m$ sets $A_1,…,A_m \subset \mathbb{R}^n$ if there exist isometries $\phi_1,…,\phi_m:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that: ...
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5answers
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Intuitive meaning of Limit Supremum?

I am trying to understand the difference between the following two equations: $$\bar{P} = \limsup_{t \to \infty}\frac{1}{t} \sum_{\tau = 0}^{t-1}E\{P[\tau]\} < \infty$$ and $$\bar{P} = \lim_{t ...
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5answers
536 views

Example of a ring with an infinite inclusion chain of ideals [closed]

I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)
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1answer
52 views

Difference between flow and solution of ODE

I am reading Wikipedia's entry on Flow and it is not clear the distinction between solution of an ODE and the flow of an ODE. In particular it is clearly written $φ(x_0,t) = x(t)$, then what is the ...
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1answer
30 views

Definition of a recursive ordinal

I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows: "...an ordinal $\alpha$ is said to be recursive if there is a recursive ...
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1answer
29 views

What would be a counterexample for a point which is a limit point of isolated singularities?

Let $G$ be open in $\mathbb{C}$ and $f:G\rightarrow \mathbb{C}$ be a function. Define $D$ as the set of points in $G$ at which $f$ is complex-differentiable. That is, $p\in D$ iff $\lim_{z\to p} ...
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1answer
53 views

Example of graph with specific $\chi (G)$, $\omega (G)$, $\beta (G)$

Find an original example of a graph whose chromatic number does not equal its clique number, yet whose clique partition number equals its independence number. Chromatic number: $\chi(G)$ is the ...
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18answers
1k views

Accidents of small $n$

In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that ...
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1answer
42 views

Give an example for if … [duplicate]

Prove that if $H$ is a normal subgroup of a group $G$ and has index $n$, then for any $g\in G$ we have $g^n\in H$. Give an example for if $H$ is not normal, the mentioned statement is not correct. ...
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2answers
20 views

substitutional interpretation of quantifiers: examples?

About the differences between propositional logic and (first order) predicate logic, given that if my basis is propositional logic I have to substitute the universal and existential quantifiers with ...
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2answers
46 views

Propositional logic vs predicate logic: examples?

About the difference between the propositional logic and the (first order) predicate logic-> can you give me one or more remarkable examples which underly the differences and the similarities ...
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2answers
66 views

Are there powerful ways to use the topological definition of continuity in real analysis?

In the lectures for introductory real analysis, my professor repeatedly told the class that the topological definition of continuity (preimage of open is open) is the most powerful version of ...
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4answers
106 views

Theorems Implying their Own Generalization [closed]

Are there any examples of theorems which were later found to imply their own generalization? Here's an example of what I mean: Hypothetically, suppose you proved Fermat's Little Theorem: $a^p \equiv ...
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2answers
48 views

Middle cancellation in a group

I can not find a such example, ( It's my first course in abstract algebra) Give an example of a group with elements $a,b,c,d$ and $x$ such that $axb=cxd$ but $ab\neq cd$. (Hence "middle cancellation" ...
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4answers
119 views

Prove or disprove: If $n^3$ is odd then $n$ is odd.

If $n^3$ is odd, then $n$ is odd. I need to prove or disprove by means of counterexample why this is true or false. $\forall x P(x) = x^3$, $x = 1,3,5,7,9$ I am having a very difficult time ...
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32answers
7k views

What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ...
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1answer
36 views

When checking the absolute continuity of a measure, is it enough to consider a generating algebra?

Let $(X,M)$ be a measurable space, and $M=\langle A \rangle$ in which $A$ is an algebra on $X$. Suppose that $v$ is a signed measure and $m$ is a positive measure on $(X,M)$. Now, can we say: $v$ is ...
2
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1answer
38 views

Can we construct a continuous but nowhere differentiable surface?

In my analysis course we learned about the Weierstrass function which is continuous but nowhere differentiable, is it possible to make a surface which is continuous and nowhere differentiable?
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3answers
64 views

Trivial power of line bundle

I'm trying to understand the following: Thinking of a line bundle as a bunch of locally generating sections together with transition functions (which in this case are just multiplication by local ...
234
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29answers
18k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
3
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1answer
114 views

Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either: Proofs i.e. pointwise equicontinuity is uniform ...
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2answers
26 views

An example of a prime quotien ideal where his corresponding ideal is not prime

If $R$ is a comutative ring with identity ring and $K$ is an ideal from it, let $R'=R/K$ and $I$ an ideal of $R$ satisfy $K\subseteq I$ and $I'$ is the coresponding ideal of $R'$ (we knew that ...
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2answers
476 views

Examples of Separable Spaces that are not Second-Countable

In this post I give an example of a separable space that is not second-countable. What are other good examples?
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3answers
102 views

What exactly is the maximal solution of an ODE and why do we care?

I am reading these notes on the definition of a maximal solution of an ODE i.e. http://www.math.lmu.de/~philip/publications/lectureNotes/ODE.pdf But the definition is so abstract and no example is ...
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4answers
295 views

Examples of Monads and their Algebras

I'd like to get some examples of monads; specifically, I'd love a big list of different monads and a description of what their algebras are. Alternatively, online resources and especially exercices ...
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2answers
1k views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
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0answers
212 views

Example of integral domain with infinitely ascending chain of ideals. [duplicate]

I am looking for an integral domain in which we have an infinitely ascending chain of ideals. Clearly, this can't be a PID. Also, I am looking for examples other than infinite dimensional fields, ...
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1answer
31 views

An example of smooth compactly supported function with non-vanishing Fourier transform

I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the ...
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0answers
37 views

Greedy algorithm fails to give chromatic number

Produce a graph and degree sequence for which the greedy algorithm fails to give the chromatic number. My first example is below- The first labeling uses 2 colors which is the chromatic number and ...
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1answer
49 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
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532 views

A Compact Hausdorff Space with no Manifold Structure? [closed]

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
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1answer
36 views

Example of maximum modulus principle

As it's known , an holomorphic($\neq constant$) function $f:G\subseteq\mathbb{C}\rightarrow \mathbb{C}$ has maximum modulus on $\partial G$ . I wuold an example of a function holomorphic on a disk ...