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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3
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0answers
35 views

Example of a complex manifold with certain curvature properties?

Are there (nice) examples of a complex manifold such that the sectional curvatures through all the complex planes are non-positive but the sectional curvatures through the real planes are mixed?
2
votes
3answers
1k views

Proof by contradiction using counterexample

Why can't we use one counterexample as the contradiction to the contradicting statement? Example: Let a statement be A where a-->b. We can prove A is not true by finding a counter example. Now, in ...
1
vote
0answers
18 views

Minimum edge cover of the Petersen graph

I want to find a minimum edge-cover of the Petersen graph. It is my understand that an edge cover is a set of vertices which is connected to all edges in the graph. Is this correct? I'm struggling to ...
42
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10answers
12k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
2
votes
1answer
58 views

Example of a ring $R$ such that $R\otimes_R R\not\simeq R$

I want to show the following statement: If a ring $R$ is commutative and $I,J\triangleleft R$, then $$ R/I\otimes_R R/J\simeq R/(I+J). $$ I can easily show this, assuming that $1\in R$. What ...
3
votes
1answer
58 views

If $x$ is even and $y$ is odd, then $x+y$ is even.

I was also asked to proof if I say the above statement is true and give a counter example if I say it is False. Moreover, I prefer the statement to be false because the sum of any even and odd number ...
0
votes
2answers
38 views

Convolutions with $L^\infty$ functions

I read the following theorem about convolutions with $L^p$ functions in real analysis: Let ${\phi_n \in C^\infty_c({\bf R}^d)}$ be a sequence of approximations to the identity. If ${f \in L^p({\bf ...
3
votes
1answer
136 views

Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them. But I am sure the question is correct
3
votes
1answer
278 views

$\sum a_n$ divergent with $\lim(n a_n)=0$

Can we find an example of a divergent series $$\sum_{n=1}^\infty a_n$$ where the sequence $(a_n)$ is a decreasing sequence of real numbers, but such that $$\lim_{n\to\infty}(n a_n)=0$$
0
votes
1answer
142 views

Modifications of the necessary condition for series convergence [closed]

Could you help me prove the two following lemmas on series convergence? 1) Prove that if $\sum_{n=1} ^{\infty} a_n$ is a series of positive real numbers convergent to $0$, where $(a_n)$ is a monotone ...
4
votes
3answers
39 views

Rate of convergence of summable sequence

Suppose $a_n$ is a nonnegative real sequence such that \begin{equation} \sum_n a_n <\infty. \end{equation} What do we know about $a_n$? We know $a_n\to 0$. We know $$\lim\inf n a_n = 0.$$ But can ...
10
votes
3answers
804 views

M compact $p\in M$ , there exist $f:M-p\to M-p$ continuous bijection but not homeomorphism?

Let M be a compact metric space. We know that if $ g:M\to M$ is a continuous bijection then it's a homeomorphism. But I want to know, if I have a continuous bijection $ f:M - \left\{ p \right\} \to M ...
24
votes
6answers
8k views

Functions which are Continuous, but not Bicontinuous

What are some examples of functions which are continuous, but whose inverse is not continuous? nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry.
7
votes
2answers
621 views

Example of a continuous function with a discontinuous inverse

What is an example of a function $f: \Bbb R^n \rightarrow \Bbb R^m$ such that $f$ is continuous and injective but that $f^{-1}$ is not continuous. Our professor teased us with the notion but I ...
1
vote
1answer
31 views

Restriction of an isomorphism to an invariant subspace may fail to be surjective

I'm wondering whether the restriction of a vector space automorphism $f : V \to V$ to an invariant subspace $W \subset V$ can fail to be surjective, i.e. $f\vert_W : W \to W$ is not an ...
1
vote
1answer
23 views

A positive square integrable random variable whit non square integrable inverse

I'm looking for an example of a Square Integrable Random Variable, whose multiplicative inverse is not Square Integrable.
4
votes
1answer
103 views

Two points of view on constructible sets

This question is aimed at understanding the relationship between two different definitions of the constructible sets in a Noetherian scheme, both of which I encountered in Atiyah-MacDonald's ...
1
vote
1answer
32 views

Counterexample for uniform integrability of an $\mathbb{L}^1$-bounded sequence

I need to find an example such that $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|] < \infty$ but $\{X_n\}_{n \ge 1}$ are not uniformly integrable. I can show that if $\{X_n\}_{n \ge 1}$ are uniformly ...
2
votes
2answers
62 views

An example of ideal $I$ such that $I^{ec}\neq I$

Let $A$ be a commutative ring, $S \subseteq A$ a multiplicative system and $i_S : A \rightarrow S^{-1}A$ the canonical morphism. Can you give me an example of ideal $I \unlhd A$ such that $I^{ec}\neq ...
1
vote
1answer
34 views

A uniformly integrable sequence that is not uniformly integrable in $\mathbb{L}^p$ for $p>1$

I need to construct an example to show that if for the sequence $\{X_n\}_{n \ge 1}$ we have that $\mathbb{E}\left[\sup\limits_{n \ge 1} |X_n|\right] < \infty$, the sequence in not necessarily ...
3
votes
1answer
40 views

Example for uniformly integrable $\mathbb{L}^2$-bounded sequences

How can I construct an example to show that the sequence $\{X_n\}_{n \ge 1}$ can be $\mathbb{L}^2$-bounded, however it has $\mathbb{E}\left[\sup\limits_{n\ge 1} |X_n|\right]=\infty$. Basically I need ...
0
votes
0answers
43 views

Linear Algebra, linear transformation

I was asked to give an example that satisfies the properties of a linear transformation, but that is not a function. Any help please? The question is from my teacher, and he insists that there ...
3
votes
1answer
79 views

If $\min(\alpha,F)$ has only one root in $E$, must $\min(p(\alpha),F)$ have only one root in $E$

Let $F<E$ be an algebraic field extension. Let $\alpha\in E$ be such that $\min(\alpha,F)$ has only one root in $E$ (which will be $\alpha$).Is it true that for any $p(x)\in F[x]$ we must have: ...
-2
votes
2answers
70 views

'Non-example' to the Lagrange theorem. [duplicate]

Lagrange's theorem: Let G be a finite group and let H be a subgroup of G. Then, $|H| | |G|$ The converse does not hold in general. 'Non-example': $G=A_{4}$ where $A_{4}$ is the alternating ...
6
votes
1answer
57 views

Can $f_n\to f$ uniformly, $f'_n\to g$ uniformly, but $f$ not being differentiable?

Just the question in the title, I know that if $f_n$ are differentiable, $f_n\to f$ uniformly, $f'_n\to g$ uniformly and $f$ is differentiable, then $f'=g$, so I'm looking for a counterexample if we ...
16
votes
1answer
9k views

Lebesgue measurable set that is not a Borel measurable set

exact duplicate of Lebesgue measurable but not Borel measurable BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck... In short: Is there a Lebesgue ...
2
votes
0answers
47 views

Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the ...
7
votes
2answers
2k views

Can someone explain and give brief examples of centralizer and normalizer?

This is a chapter on group action and I need to better understand its relationship with centralizer and normalizer. Quick examples would be very helpful! Thank you so much.
4
votes
2answers
54 views

Group Operations/ Group Actions

I'm currently taking my first abstract algebra course and am learning about group actions, orbits, and stabilizers. I'm reading the Artin textbook and I am not very clear of what exactly a group ...
2
votes
3answers
180 views

Examples of Non-Faithful Group Actions

I cannot find anywhere a relatively simple example of a non-faithful group action. I feel I understand the definition relatively well, however I can't come up with any ideas for one in my head (and ...
3
votes
3answers
492 views

Concrete examples of group actions.

First, a little motivation: I have read the section on Group Actions in Dummit & Foote, the wikipedia page, and (countably many) other references. And seemingly without exception, they only offer ...
1
vote
2answers
195 views

An abelian Banach algebra without characters

Can one give an example of an abelian Banach algebra with empty character space? Such algebra must be necessarily non-unital. I couldn't find any examples of such algebras. Thanks!
2
votes
4answers
45 views

Numerical property $P(n)$ such that $\forall n P(n)$ is false but a counterexample is difficult to find

I would like to find a nontrivial property $P(n)$ for $n \in \mathbb N$ such that $\forall n P(n)$ is false but the first counterexample can be found only for "very high" $n$ (so high that it wouldn't ...
0
votes
2answers
55 views

Unnatural homomorphism form domain $R$ to $Frac (R)$

There is a natural homomorphism for $R$ to $Frac (R)$ that sends $r\rightarrow(r,1)$, but beside this injective homomorphism, is there example of ring $R$ s.t there exist another injective ...
2
votes
3answers
81 views

I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy

I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy ? I tried to find such sequence $x(n)=1/2,1/3,1/2,1/3,1/4,1/2,1/3,1/4,1/5,,,,$ it's not Cauchy since it is ...
2
votes
5answers
101 views

If $\{x_n\}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?

Since the definition of Cauchy sequence is: Understanding the definition of Cauchy sequence, I noticed we need an absolute value for $a_m-a_n$ in the definition so the statement would be false. But I ...
10
votes
1answer
148 views

What are some interesting counterexamples given by finite topological spaces?

According to Wikipedia, 'finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures.' I have been studying the book ...
1
vote
0answers
29 views

Two alleged counterexamples (about boolean algebras)

Trying to solve this question, I propose two possible counter-examples. Please help me to understand whether these cases are really counter-examples. Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) ...
7
votes
3answers
3k views

Finding number of matrices whose square is the identity matrix

how can we find the number of matrices with real entries of size $9 \times 9$ (up to similarity) such that $A^{2}=I$? I first thought about the following: Notice $A$ satisfies the polynomial ...
10
votes
5answers
12k views

If $A^2 = I$ (Identity Matrix) then $A = \pm I$

So I'm studying linear algebra and one of the self-study exercises has a set of true or false questions. One of the question is this: If $A^2 = I$ (Identity Matrix) Then $A = \pm I$ ? I'm pretty ...
1
vote
1answer
52 views

Show that every nearly compact space is almost compact space but the converse is not true

I am learning about the almost compact space and nearly compact space. I know that every nearly compact space is almost compact space but the converse is not true in general. So i need an example of ...
1
vote
5answers
573 views

Non-Metrizable Topological Spaces

What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. ...
0
votes
1answer
24 views

Counter example of Algebraic sets

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
213
votes
29answers
20k 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 ...
3
votes
1answer
50 views

Does equal cardinality and one set being a subset of the other prove equality? [closed]

I'm currently solving a quite specific problem and in the final step I made a statement that can be generalised such that: $$((|A|=|B|)\wedge(A\subset B)) \implies (A=B)$$ Whilst this is clearly ...
1
vote
2answers
63 views

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. ...
1
vote
2answers
86 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 ...
2
votes
4answers
249 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 ...
14
votes
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 ...
1
vote
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 ...