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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Give the example of compact set with infinite countable derived set

Can anyone give me an example of compact set of which the derived set is infinitely countable set?? thks in advance, I have no idea about this .
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Image is not a manifold when considered as a subset: how is this possible?

Wikipedia offers two definitions of a submanifold: One is that it is the image of an immersion. But I can't make sense of the remark that " in general this image will not be a submanifold as a ...
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2answers
51 views

Categories with some but not all exponentials

The introductory examples typically given of exponential objects in categories in fact involve categories which have all exponentials. Are there not-too-esoteric examples of categories of ...
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7answers
1k views

Right English wording for “counterexamples to a theorem”

This question is about the right English wording. I give here what I call "counterexamples to Banach fixed-point theorem". What I do, is that I look to what happen if some hypothesis of the theorem ...
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1answer
23 views

Is a Noetherian normal local domain (universally) catenary?

Let $R$ be a ring. Then $R$ is $\textit{catenary}$ if for a pair of prime ideal $p \subseteq q$, all maximal chains of prime ideals $p = p_0 \subseteq p_1 \subseteq \dots \subseteq p_n = q$ have the ...
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4answers
58 views

Why in general there is no systematic way to find counterexamples? What kind of property do they all break that lead to this? and other things

We came across counterexamples in many areas of mathematics: For example Sum of irrational numbers not necessary being irrational The "Windmill blade" function (for lack of a better name of one of ...
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2answers
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A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
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1answer
34 views

looking for example of infinite p-group of nilpotency class 2

Is there infinite p-group of nilpotency class 2? If p=2 or p=3 would be better. and I prefer simple examples
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1answer
22 views

An example of open closed continuous image of $T_2$-space that is not $T_2$

Engelking in his "General Topology" states that $T_2$ separation axiom is not preserved under open closed continuous surjections. In "General Topology" by Stephen Willard I have found two separate ...
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3answers
479 views

sequentially continuous on a non first-countable

Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this ...
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1answer
50 views

Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
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2answers
43 views

An example of open closed continuous image of $T_0$-space that is not $T_0$

Engelking in his "General Topology" states that $T_0$ separation axiom is not preserved under open closed continuous maps. But I can't find any example of open closed continuous image of $T_0$-space ...
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2answers
103 views

An example of a group G that satisfies $x^2 =e$ , for all $x \in G$

I thought for a long time about which group will have this property but I didn't get it. Is this a valid one so if G = {a,b} then ab = b and b is the identity for example then aa = b , ab = b and ...
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1answer
57 views

need examples of different groups

I need example of different groups having different properties like: class 2 or 3 cyclic commutator cyclic center $Z(G)\le \Phi(G)$ redei group $G=\langle aG',bG' \rangle $ and ... Is there books or ...
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5answers
59 views

Countable/Uncountable collections

I'm asked to produce an example of a countable collection of disjoint open intervals. At first I had trouble seeing how this is possible since open intervals are not countable. My idea is to have ...
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2answers
12k views

Examples of Simpson's Paradox

I'm looking for fresh examples of Simpson's paradox for use in my statistics courses. The examples I've been using are fine, but I'd like to have some new ones, and I'm hoping folks here might know a ...
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1answer
42 views

Jensen's inequality problem [closed]

I want to know an example of a infinite measure space $(\Omega, \mathcal{F},\mu)$, real valued function $g$ and convex function $\phi$ defined on the real line s.t. $$\phi\left(\int g d\mu\right) > ...
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1answer
25 views

Example of a pointwise convergent functional sequence that is not compactly convergent.

I'm looking for an example of a pointwisely convergent functional sequence $\{f_n\}_{n \geq 0}$ (where $f_n:\mathbb{R}\to\mathbb{R}$) that is not compactly convergent. I'm not sure if it is even ...
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1answer
64 views

Why is it so difficult to obtain the spectral properties related to infinite matrices, especially when they are not symmetric?

I believe most of the spectral theory is revolving around the bounded self-adjoint linear operators being analogous to real symmetric infinite matrices. Whereas, there are cases when the matrices are ...
3
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2answers
62 views

Is the minimum of a parametric convex function convex again?

Let $I$ and $J$ be compact intervals. Let $f:I\times J\to\mathbb R$ be differentiable and strictly convex. Is the function $g:I\to\mathbb R$ defined by $$ g(x) = \min_{y\in J} f(x,y) $$ ...
2
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0answers
34 views

What is an illustrative example of a Finsler manifold?

I've attempted to get a bead on Finsler manifolds before attending an upcoming seminar that involves them. I've done some reading in the literature and think I understand that they are similar to ...
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1answer
58 views

Example of a function in $L^2(\mathbb{R})$ with derivative not in $L^2(\mathbb{R})$.

We know examples of a function which doesn't lie in $L^2(\mathbb{R})$ with derivatives in $L^2(\mathbb{R})$: $$f_1(x) = \mathrm{arctg}(x) \notin L^2(\mathbb{R}), \qquad f_1'(x) = \frac{1}{x^2+1}\in ...
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2answers
343 views

Is every subgroup of a normal subgroup normal?

Is every subgroup of a normal subgroup normal ? That is if $H$ is a normal subgroup of a group $G$ and $K$ is a subgroup of $H$, then $K$ is a normal subgroup of $G$. Is it true ? If not what is the ...
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4answers
244 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
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55 views

Simpler version of dogbone space construction

In "The cartesian product of a certain nonmanifold and a line is $E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold, $B$, such that $B\times \Bbb ...
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1answer
24 views

Where can I find the nowhere subdifferentiable example of rockafellar?

I'm told that Rockafellar gave an example of a real extended function defined on a locally convex space, whose subdifferential is empty at each point of its domain. The function is proper, lower ...
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6answers
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Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
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1answer
37 views

Confusion regarding the $\omega$-limit of a set in a flow

In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot ...
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2answers
31 views

What is an example of Gâteaux differentiable but not Fréchet differentiable at a point in a finite-dimensional space?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$ such that $V$ is finite-dimensional. Let $E$ open in $\mathbb{K}$ and $p\in E$. Let $f:E\rightarrow W$ be Gâteaux-differentiable at $p$. Is $f$ ...
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1answer
226 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
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Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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1answer
20 views

Question on existential quantifier.

Let us consider the following predicates. $A(x)$: $x$ is $A$ type. $B(x)$: $x$ is $B$ type. Then convert the following statement in terms of predicate expression. Some $A$'s are $B$. Then which ...
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3answers
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Counterexample to “$A \to B, A \to C$, therefore $B \to C$”

We have $A\to B$ and $A\to C$. I need counter-examples to: '$\therefore B\to C$'. More formally, disprove: $$ (A\to B)\land(A\to C)\to (B\to C)$$ I have $A$ is a blackbird, $B$ is 'is black', $C$ ...
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5answers
151 views

An Example of a Nested Decreasing Sequence of Bounded Closed Sets with Empty Intersection

Could someone provide me with an example of a metric space having a nested decreasing sequence of bounded closed sets with empty intersection? I first thought of Cantor set but the intersection is not ...
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9answers
936 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 ...
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2answers
29 views

Additive group with out element 0 and generator should n't be 1

The additive group $\mathbb{Z}_n$ for any natural number $n$ forms an additive group which has 0 as identity element and 1 as generator. Please provide me few examples of additive groups (with ...
3
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0answers
65 views

Find two homeomorphic topological spaces and a bijective continuous map between them which is not homeomorphism.

I'm aware that it is duplicate, but I'd like to know whether my example is appropriate or not. Let our function $f$ be on the set $\mathbb{Q}\cap\mathbb{Z}$ induced by standard topology of a line. ...
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2answers
423 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
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0answers
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Examples of Riesz homommorphisms and solid linear subspaces [closed]

Hi I'm looking for some standard examples of Riesz homomorphisms and solid linear subspaces.
12
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9answers
716 views

What is the difference between dense and closed sets?

I am self studying topology and is a bit stuck on the difference between dense and closed sets. Intuitively, a dense set is a set where all elements are close to each other and a closed set is a set ...
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6answers
115 views

Operator which is symmetric but not associative?

Addition and multiplication are symmetric and associative. But I have no idea about operators which are symmetric but not associative. Please help me listing any such 2-arity operators.
3
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1answer
59 views

Is there a generalization of Jordan curve theorem? Not in higher dimension, but in the plane?

Jordan curve theorem (bit generalized one) Let $C_1$ and $C_2$ be closed connected subsets of $S^2$ whose intersection consists of two points. If neither $C_1$ nor $C_2$ separates $S^2$, then ...
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1answer
59 views

Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
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0answers
29 views

Just use the expected value for the random coefficient in a differential equation

We often encounter differential equations with some coefficients that are random variables. One way to solve these problems is to replace the random coefficient with its expected value (EV). Then we ...
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1answer
25 views

Is this a Darboux function?

Let $f(x)=x$ if $0\leq x\leq 1$ and $f(x)=x-\frac{1}{2}$ if $1<x\leq 2$. This is a discontinuous function on $[0;2]$ but it satisfies the intermediate value theorem so it's a Darboux function. ...
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2answers
559 views

Why doesn't logic, math, physics etc have a symbol for “example”?

We have symbols for everything but there is no symbol for "example" despite examples being fundamental to achievements. Why is there no symbol for "example" when there are symbols for everything ...
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1answer
51 views

Example of $f:\mathbb{R}\to\mathbb{R}$ injective and bounded, but with inverse not bounded or injective.

I am trying to come up with an example of a bounded and injective function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}$ is not injective or bounded. What are examples that could apply in this ...
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0answers
25 views

A modified Shapiro's Tauberian theorem? Proof or counterexample

Let $\{a(n)\}$ be a nonnegative sequence such that $$\sum_{n\leq x}a(n)[x/n^{2}]=x^{2}\log x + O(x^{2})$$ for all $x\geq 1$, where $[y]$ denotes the greatest integer $\leq y$. Is true that the ...
2
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2answers
93 views

Examples of a cayley table that represents a structure that satisfies all group axioms except associativity

I'm curious if there are any cayley tables on a finite amount of elements that satisfy the axioms of a) closure, b) identity, and c) inverse, but that for at least one triple of elements do not ...
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2answers
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Examples of types of mathematical models

I am a student currently doing a course on modelling and simulation. I came across the classifications of mathematical models and studied that they can classified as static or dynamic, deterministic ...