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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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
45 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
287 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 $\{f(...
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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
68 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 $1,...
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0answers
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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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25 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
545 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
53 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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31 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 well-...
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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} (f(z)...
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1answer
61 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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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
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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
51 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
140 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
112 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
58 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
123 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
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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
38 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 ...
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1answer
39 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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68 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 ...
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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 ...
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1answer
122 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
553 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
104 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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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 on ...
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2answers
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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
213 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
36 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
49 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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5answers
543 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
40 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 ...
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5answers
7k views

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
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3answers
87 views

Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it? Though there are ...
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3answers
98 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
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3answers
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Is a bounded and continuous function uniformly continuous?

$f\colon(-1,1)\rightarrow \mathbb{R}$ is bounded and continuous does it mean that $f$ is uniformly continuous? Well, $f(x)=x\sin(1/x)$ does the job for counterexample? Please help!
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2answers
216 views

An example of a bounded, continuous function on $(0,1)$ that is not uniformly continuous

I can not find the example of a continuous function on $(0,1)$ that is bounded on $(0,1)$, but not uniformly continuous on $(0,1)$. Is there any? Thank you.
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Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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5answers
286 views

How to find a differentiable function with bounded derivative satisfying some boundary conditions?

I am trying to find an example, preferably an explicit one, of a differentiable function $g:\mathbb{R}\rightarrow \mathbb{R}$ satisfying the following conditions: $\displaystyle g(0)=0, g(1)=1, g(-1)...
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1answer
161 views

Examples of decreasing sequences of closed sets with constant diameter and empty intersection in complete metric spaces

Looking through older exams from the topology class I'm taking, I found an interesting problem. Give an example: $ (X, d) $ - a complete metric space $ F_1 \subset F_2 \subset F_3 \subset ... $ - a ...
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1answer
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An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and $\text{diam}(...
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5answers
2k views

Counter-examples of homeomorphism

Briefly speaking, we know that a map $f$ between $2$ topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous. So, can anyone give me $2$ counter ...
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3answers
82 views

Example of Topological Group

I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology. ...
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1answer
51 views

Natural examples of multiplicatively-graded rings

A ring $A$ is multiplicatively graded if, as an additive group, it decomposes as $$A=\bigoplus_{n\geq 1} A_n$$ and the product satisfies $A_n \times A_m \rightarrow A_{nm}$. A natural example of ...