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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1answer
48 views

Analouge of the Mean value theorem for holomorphic functions

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be entire. Let $w_1,w_2$ be any two distinct complex numbers. Must there exist $c\in \overline{B_{|w_2-w_1|}(w_1)}$ such that ...
3
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1answer
77 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
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0answers
18 views

Is a extension of a premeasure preserves outer-measure generated by the premeasure?

I have proved the follow: Let $X$ be a set. Let $S$ be a semi-ring of subsets of $X$. Let $\mu$ be a premeasure on $S$. Let $\overline{\mu}$ be a premeasure on a ring generated by ...
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0answers
108 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
10
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1answer
242 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
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2answers
43 views

Class of algebras defined by quasi-identities is not closed under taking the quotients

Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By quasi-identity I mean the formula of the form $$(\forall x_1) (\forall x_2) \dots (\forall x_n) ...
3
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3answers
138 views

What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
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0answers
25 views

A question on the classical Mrowka space

Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset ...
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2answers
101 views

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$? [closed]

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$ ? Im sorry I am confused about ring theory. Its all new to me.
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2answers
50 views

Examples of ergodic geodesic flow

Are there any good examples of a geodesic flow that is ergodic? I know the result that states that the geodesic flow for manifolds with negative curvature are ergodic, but I'm fishing for some ...
11
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1answer
179 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
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9answers
500 views

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
4
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1answer
73 views

The function $f(x)=(x\vee a)\wedge b$ in a lattice.

Is there an algebraic modular lattice $(X,\vee,\wedge)$ and $a,b\in X$ with $a\le b$ such that the function $$f:X\to X$$ $$f(x)=(x\vee a)\wedge b$$ is not $\vee$-homomorphism?
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1answer
68 views

Is every Hilbert space an $L^2$ space

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
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0answers
59 views

Maximum measurable collection and Caratheodory extension

Remember Caratheodory extension theorem: we get an outer measure by extending a premeasure over an algebra. We also have Caratheodory theorem, which give us a measure by restricting that outer measure ...
1
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1answer
50 views

Counter example for absolutely continuous measure

I need a example for the following statement: "Given a pair of finite measures $(\mu,\nu)$ on a given measurable space $(\Omega, \mathbb{A})$ is said to have property $P$ if for every $\epsilon >0$ ...
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2answers
73 views

Differentiability of a certain piecewise function

Consider the function $$ f(x)=\begin{cases} x & \textrm{if } x \textrm{ is rational} \\ -x & \textrm{if } x \textrm{ is irrational} \end{cases} $$ It is well-known that $f(x)$ is continuous at ...
5
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1answer
145 views

Examples of Baire class 2 functions

Do you know of examples of Baire class 2 functions which are not Baire class 1 functions, besides the the indicator function of the rationals and the indicator function of the Cantor set?
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2answers
174 views

Examples of nowhere continuous functions

Do you know examples of nowhere continuous functions, besides the Dirichlet function and its modifications?
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1answer
13 views

A criterion for complete lattice.

Is there an infinite partially ordered set $(X,\le)$, in which for each $A\subseteq X$, either $\inf A$ or $\sup A$ exists but for some $A\subseteq X$ either $\inf A$ or $\sup A$ does not exist.
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2answers
142 views

Counterexample: multiplying modules by elements of an ideal vs. taking linear combinations

Let $R$ be a ring (commutative, unital) and $M$ an $R$-module. Let $I \subset R$ be an ideal. We make the following definitions: $$ A := \{ am \ | \ a \in I,\ m \in M \} $$ $$ B := \left\{ ...
2
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0answers
46 views

Restricted continuity implies continuity

When teaching calculus, we instruct students to calculate multivariate limits using the following theorem: If $\gamma$ is a smooth curve with $\gamma(0) = a \in \newcommand{\R}{\mathbb{R}}\R^n$ ...
8
votes
2answers
71 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
3
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1answer
79 views

Does $\phi\circ g$ convex imply $\phi\circ f\circ g$ convex?

Assume that $\phi$ is an increasing function and $g$ is a decreasing function with $\phi\circ g$ convex. If $f$ is an increasing convex function does this imply $\phi\circ f\circ g$ is a convex ...
5
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1answer
75 views

Uniquely complemented lattice that is non-modular

I'm looking for an explicit example of a uniquely complemented lattice that is non-modular, since neither of the two non-modular lattices described here at wikipedia have this property. Thanks.
2
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1answer
49 views

Example of a function between boolean lattices that preserves $(\top,\bot,\wedge,\vee)$ but not complements.

Its easy to find boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ such that $f$ preserves both top and bottom elements, as well as binary meets, but not complements. ...
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2answers
187 views

Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent

$p>1$ is a integer, Show a convergent series $\sum\limits_{n=1}^\infty a_n$, $a_n\in\Bbb R$, such that the series $$\sum_{n=1}^\infty a_n^p$$ is divergent p.s. If $p>1$ is not an integer ...
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2answers
42 views

Significance of low order terms in base expansion of integer square root

My head is turning into a uniform gel of random thoughts! I cannot see a proof or find a counterexample to the following: Conjecture: Let integer $x$ be expressed as $a_3 \, b^3 + a_2 \, b^2 + a_1 \, ...
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0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
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4answers
90 views

Looking for a counter example - Normal subgroups and quotient group

I need to find a counter example for the following: Let $G$ be a group, and let $A,B\triangleleft G$ be two normal subgroups of $G$. if $G/A\cong B$ then $G/B\cong A$. Here are my thoughts so far: ...
5
votes
1answer
47 views

$L^1(\mathbb{R}^n)$ functions not in $\mathcal{H}^1(\mathbb{R}^n)$

I am wondering how to imagine the Hardy space on $\mathcal{H}^1(\mathbb{R}^n)$ and in particular what sort of functions are in $L^1(\mathbb{R}^n)\backslash\mathcal{H}^1(\mathbb{R}^n)$. Furthermore, is ...
1
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2answers
46 views

Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
9
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4answers
708 views

Example of a finite non-commutative ring without a unity

Give an example of a finite, non-commutative ring, which does not have a unity. I can't think of any thing which fits this question. I was thinking $M_2(\mathbb{R})$ but it has the identity. Any ...
2
votes
3answers
53 views

Non-trivial order on a set with infinitely many minimals/maximals

I am looking for an example of each: 1) A non-trivial order on a set A such that there are infinitely many minimal elements and 2) A non-trivial order on a set A such that there are infinitely many ...
1
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3answers
57 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
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1answer
46 views

The Tangent Disc Topology is developable

A well-known example of a Moore space is the Tangent Disc Topology. I want to show that the Tangent Disc Topology is a developable space, i.e. it has a development. But I could not find the proof of ...
2
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1answer
52 views

Does there exist a suitable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n^m) = mf(n)$? I think no.

Any ideas how to prove that no injection $f : \mathbb{N} \rightarrow \mathbb{N}$ whose image is closed under multiplication by elements of $\mathbb{N}$ satisfies the following identity? $$f(n^m) = ...
1
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1answer
38 views

An isometrie $\varphi: S_1\to S_2$ which cannot be extended into distance-preserving map

I'm searching for an example isometrie $\varphi: S_1\to S_2$($S_i$ are regular surfaces) which cannot be extended into distance-preserving maps $F: \Bbb R^3 \to \Bbb R^3$. A reference or hint will ...
6
votes
2answers
115 views

Do silly-rings exist?

A ring can be defined as a near-ring satisfying two-sided distributivity, whose underlying additive group is Abelian. Negating this second stipulation, we obtain the following definition. A silly-ring ...
6
votes
3answers
92 views

Structures with addition, multiplication and exponentiation.

The set $\mathbb{N}$ can be viewed as a mathematical structure with operations off addition, multiplication and exponentiation. Observe that: It forms an Abelian monoid under both addition and ...
3
votes
1answer
38 views

Are there interesting examples of medial non-commutative semigroups?

There exist semigroups $S$ (written additively) such that $S$ is medial, meaning $(a+b)+(a'+b') = (a+a')+(b+b')$. $S$ is not commutative. Example. The left (and right) zero semigroups are all ...
2
votes
1answer
178 views

Existence and uniqueness of limit of inverse function

Let $f:(a,b) \rightarrow \mathbb{R}$ be a one to one function. If $x_0$ is a point of the open interval $(a,b)$ such that $\lim_{x \rightarrow x_0} f(x) = l$, is it necessary that $\lim_{x \rightarrow ...
4
votes
1answer
264 views

Counterexample for the Open Mapping Theorem

I would like to ask a counterexample for the open mapping theorem: Find a discontinuous linear mapping $T:X \to Y$ such that $T(X)=Y$ and $X,\;Y$ are Banach but $T$ is not open. Could you help me ...
1
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2answers
58 views

Probability of result

Morning, A inbound contact centre says they will answer calls within 3 mins 70% of the time, if I ring the call centre 5 times over a week what is the probability I get my calls answered in under 3 ...
2
votes
2answers
50 views

Continuous bijection whose inverse is not continuous at uncountably many points

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible ...
1
vote
1answer
29 views

Let $f\geq 1$, Is the function $p\rightarrow \int |f|^p d\mu$ continuous

Let $f:X\rightarrow [0,\infty[$ be a measurable function that is greater than or equal to $1$ for every $x\in X$ and $\mu$ be a positive measure on $X$. Consider the function $g:]0,\infty[\rightarrow ...
2
votes
1answer
34 views

Examples of rational equivalence.

I have to do a lesson (1 h.) about a very basic introduction to intersection theory. In order to do this I'd like to find a way in order to examplain the concept of rational equivalence giving a lot ...
1
vote
1answer
48 views

Closed sets, boundary, topology.

Let A be a closed subset of the real numbers. It is always possible to find a subset B of the real numbers such that A is equal to the boundary of B? Prove if true, find a counterexample if not. I ...
2
votes
2answers
200 views

Non-measurable set in product $\sigma$-algebra s.t. every section is measurable.

Let $\Omega$ and $\Gamma$ be two nonempty sets and $\mathscr{A}$ and $\mathscr{B}$ be $\sigma$-algebras over $\Omega$ and $\Gamma$, respectively. The product $\sigma$-algebra of $\mathscr{A}$ and ...
2
votes
1answer
42 views

Submonoids of $\mathbb{N}^k$

Do you know if all submonoids of $\mathbb{N}^k$ are finitely generated? If not, can you give me a counter-example? EDIT : I mean $\mathbb{N}^k$ as a submonoid of $(\mathbb{Z}^k,+)$. I already know ...