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 ...

learn more… | top users | synonyms (2)

25
votes
6answers
9k 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
717 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 haven'...
1
vote
1answer
34 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 automorphism. ...
1
vote
1answer
27 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
107 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
40 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 ...
4
votes
2answers
82 views

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

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
43 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
49 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 ...
3
votes
1answer
82 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
76 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
58 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 ...
17
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
49 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
64 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
220 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
542 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 ...
2
votes
2answers
205 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
49 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 ...
1
vote
2answers
57 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
115 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
167 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
32 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 $f(t)=t^...
12
votes
5answers
13k 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
55 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
665 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
28 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 ...
215
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 don't ...
3
votes
1answer
64 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
66 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
92 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
276 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
388 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
51 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 ...
1
vote
4answers
50 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 L(...
2
votes
2answers
45 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} f'...
0
votes
0answers
22 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 ...
1
vote
1answer
85 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 ...
1
vote
0answers
36 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 $A$...
2
votes
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 & ...
1
vote
1answer
67 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 $l^2$ : $...
1
vote
0answers
54 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 ...
1
vote
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 ...
9
votes
3answers
292 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(...
0
votes
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 ...
2
votes
3answers
72 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,...
0
votes
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 ...