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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55 views

Is the axiom of choice really all that important?

According to this book: The Axiom of Choice is the most controversial axiom in the entire history of mathematics. Yet it remains a crucial assumption not only in set theory but equally in modern ...
5
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1answer
55 views

If all the numbers $(1^\alpha,\,2^\alpha,\,3^\alpha,\,\dotsc)$ are integer, then $\alpha$ is an integer.

A theorem of Siegel asserts that If $\beta>0$ and $2^\beta,\,3^\beta,\,5^\beta$ are integers, then $\beta$ is an integer. The following result is a beautiful consequence of this theorem ...
2
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1answer
42 views

What would be an example of free module such that cardinality of linearly independent set is greater than the rank?

Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has the IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=\mathrm{rank}(M)$. Let ...
0
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1answer
15 views

Polynomial in $\mathbb{Z}_2[x]$ that is reducible but has no roots a prime $p$ for which $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$

First, I am suppose to find a prime $p\geq 4$ where $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$. Second, I am supposed to find a fifth degree polynomial in $\mathbb{Z}_2[x]$ that is reducible ...
2
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1answer
21 views

Integral domains examples

I am supposed to give an example of 1) an infinite integral domain of characteristic $5$, and 2) an integral domain which is not a field. Respectively, examples I chose were $\mathbb{Z}_5$ and ...
1
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1answer
16 views
1
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1answer
34 views

What are other examples of complex associative operators besides, x + y +rxy, rxy, and x + y + 1/r?

I have been having fun (and frustration) in finding complex associative operators over the complex numbers. So far, I have found the 3 listed in the title (r is a constant), and also know about ...
4
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1answer
295 views

An explicit example of a differentiable function taking rational values at rational points but whose derivative is irrational at rational points

Construct an example of a differentiable function such that $$ \forall r \in {\Bbb Q}\quad f(r) \in {\Bbb Q}\text{ but } f'(r) \notin {\Bbb Q} $$ this example is not trivial, in a paper they ...
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0answers
21 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
2
votes
1answer
42 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
7
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1answer
99 views

Example of continuous function whose Fourier series doesn't converge on an uncountable dense set.

According to a well-known theorem (Theorem 5.12 in Rudin's Real and Complex Analysis), there is a dense $G_\delta$ set of continuous periodic functions $f:\mathbb{R}\to\mathbb{C}$ such that the ...
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2answers
28 views

What is the example of a not almost convergent sequence but whose Cesàro means converge?

It seems to me a sequence that is almost convergent implies that its Cesàro means converges but not vice versa. What is the example that a not almost convergent sequence whose Cesàro means converge. ...
0
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3answers
53 views

What is a counterexample to this one?

Let $R$ be a commutative ring and $A\in M_{n\times m}(R)$ where $n\neq m$. What is an example such that $\det(AA^t)\neq \det(A^tA)$? Indeed, I think it's true. If this is true, how do I prove this?
4
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0answers
27 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
4
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1answer
44 views

Zero-dimensional but not Hausdorff

Let's call a space zero-dimensional if it has a basis of clopen sets, and is $T_0$. Is there a zero-dimensional space that is not Hausdorff?
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4answers
518 views

How to come up with a counter example in linear algebra

This came up in a problem I was working on. Problem:Let $V$ be an $n$ dimensional vector space over a field $F$. Let $T:V\rightarrow V$ be a linear operator and let $W$ be a $T$ invariant subspace of ...
0
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3answers
82 views

Graph connected does not imply $f$ is continuous [closed]

Show an example of a function $\newcommand{\R}{\mathbb{R}} f: \R \times \R\to \R$ such that $f$ is not continuous, but its graph $$ \Gamma_f := \left\{\bigl((x, y), f(x, y)\bigr) \mid \text{$(x, y)$ ...
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0answers
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1
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0answers
40 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
12
votes
2answers
391 views

An example of a space which fails to be compactly generated

Does anyone know of an example of a topological space which is not compactly generated? I am using the definition in May's book "A Concise Course in Algebraic Topology." The definition is that a space ...
1
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1answer
42 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
1
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1answer
31 views

Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
184
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28answers
17k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
0
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1answer
44 views

Non-standard examples of Distributions

Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions. Additional: is there a ...
0
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1answer
12 views

What is an example of $E/F,L/E$ are normal but $L/F$ is not. [duplicate]

Let $E/F,L/E$ be normal field extensions. What would be an example such that $L/F$ is not normal?
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2answers
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What is an example of transcendental extension such that a monomorphism cannot be extended?

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$. Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism. What is an ...
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9answers
833 views

Advice on finding counterexamples

I am reaching out for specific advice on how one should go about finding counterexamples. It seems almost every time I've ever attempted a "find a counterexample" problem, I have to cheat by asking a ...
11
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4answers
195 views

Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
1
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1answer
39 views

Example of a white noise series that is not a martingale difference series with respect to its natural filtration

For a homework exercise, I am asked to find an example of a white noise series that is not a martingale difference series with respect to its natural filtration. Does anyone know an example? I read ...
8
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3answers
229 views

A strong Hausdorff condition

Is the following strong form of Hausdorff equivalent to usual Hausdorff? $X$ is strong Hausdorff if given distinct elements $x,y$ in $X$ there are open sets $U,V \subseteq X$ with $x \subseteq ...
2
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0answers
25 views

How much regularity is needed, anyway?

When doing real analysis, the difference between functions which are continuous and functions which are not is intuitive. The graph of the later may exhibit shearing, or extreme distortion (in higher ...
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3answers
61 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
0
votes
1answer
21 views

Are there an completely regular, non-Lindelöf spaces with only constant real valued functions?

Is there an example for a topological space, which is not Lindelöf, but is completely regular, on which continuous real valued functions, are, constant for all $x \in X$, or, constant for all, except ...
0
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2answers
17 views

Example of $1-1$ Correspondence with Subgroups of Factor Group

I am working out an example to deomstrate the one-one correspondence between $\{\text{subgroups of}\ D_4/N\}$ and $\{\text{subgroups of $D_4$ that contain $N$}\}$ but I am short one in $D_4$. ...
2
votes
1answer
33 views

Finding a counter example for $ \left(A+A\right)'\subseteq\left(A'+A\right)\cup\left(A'+A'\right)$

let $A'$ be the set of limit points of $A\subseteq\mathbb{R}$ and $A+B=\{x+y:x\in A,y\in B\}$, I'm required to find a counter-example for: $$ ...
6
votes
2answers
120 views

If $\frac{a+b}2$ is rational, can we say that $a,b$ are rational?

The question is if it's given that $$ {a+b\over 2} \in \Bbb Q $$ prove or disprove $a,b \in \Bbb Q$. Since it is to disprove, I tried the following method by using examples. Take $$a = 1 + \sqrt{2} ...
6
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3answers
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Reflexive space which is not uniformly convex

I found this beautiful theorem due to Milman and Pettis: Every uniformly convex Banach space is reflexive. I think it's a remarkable statement, since uniformly convexity is a geometric property ...
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1answer
16 views

Construct measures on $\sigma(B)$ that agree on $B$

Let $X=\{ 1,2,3,4\}$ and $\mathcal B=\{\{1,2 \},\{ 1,3\},\{ 2,4\},\{ 3,4\} \}$. And let $\mathscr A = \sigma(\mathcal B)$ be the $\sigma$-algebra generated by the set $\mathcal B$. I wish to construct ...
0
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1answer
49 views

Example for $a^k\equiv b^k$ and $k\equiv j$ but $a^j\not\equiv b^j\pmod n$

I need some help in the number theory please , Who can give me an example : If $$a^k≡b^k \pmod{n}$$ and $$k≡j \pmod{n}$$ is not necessary to be $$a^j≡b^j \pmod{n}$$
0
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1answer
10 views

example verification partial order

Let A= { 1, 2, 3} then R= { (1, 1), (2, 2) , (3, 3) , (1, 2), (2, 3) } ,the relation R is reflexive and anti-symmetric,i get it, but how is it following the transitive property for it to become ...
0
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1answer
19 views

Cavalieri's Principle in measure theory

The first part of Cavalieri's principle (in measure theory) states if $E$ is measurable, then almost every slice $E_x$ of $E$ is measurable. Here, it uses "almost every", so what is an example where ...
1
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1answer
963 views

Example of a Markov chain transition matrix that is not diagonalizable?

It is well-known that every detailed-balance Markov chain has a diagonalizable transition matrix. I am looking for an example of a Markov chain whose transition matrix is not diagonalizable. That is: ...
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4answers
63 views

An example of a space $X$ which doesn't embed in $\mathbb{R}^n$ for any $n$?

Apologies if this has been asked before, but couldn't find it. The definition of embedding that I'm using is this: Suppose $X$ and $Y$ are topological spaces. We call a function $f:X\rightarrow ...
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1answer
58 views

closed, convex, absorbing subset of a banach space

There is a nice theorem that every closed, convex, absorbing subset of a banach space includes an open ball arround $0$. Can you give an example where the theorem fails if we do not assume the subset ...
5
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1answer
93 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
5
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2answers
220 views

Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...
5
votes
1answer
50 views

McLain's characteristically simple group

Given a field $\mathbb{F}$, one can construct the free $\mathbb{F}$-module over the set of rational numbers. Assume that $(v_x)_{x \in \mathbb{Q}}$ is a basis of this linear space (indexed by the ...
7
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3answers
1k views

Need some help on a non-example of equicontinuity

In an attempt to better understand the definition of an equicontinuous family of continuous functions, I want to find a simple non-example. My intuition says that the family ...
165
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32answers
9k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
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1answer
17 views

Counterexample for continuous function over product topology without compactness

Suppose $f$ $(X,d_x)$: $\rightarrow$ $(Y,d_y)$ is a function between metric spaces, and $X \times Y$ has the product topology. The graph $G_f$ is the subspace $G_f$ = {$(x,f(x))$ | x $\in$ $X$}. If ...