Tagged Questions

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 ...

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.
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'...
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. ...
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.
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 ...
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 ...
82 views

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 ...
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 ...
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 ...
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 "...
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.
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 ...
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 ...
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 ...
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!
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 ...
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 ...
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 ...
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 ...
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 '...
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) ...
3k views

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 ...
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 ...
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 $A$...