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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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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0answers
13 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
2
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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
46 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
56 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
56 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 ...
0
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1answer
41 views

Jensen's inequality problem [on hold]

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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votes
1answer
24 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 ...
3
votes
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 ...
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2answers
61 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 $$ g(x) = \min_{y\in J} f(x,y) $$ ...
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0answers
33 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 ...
3
votes
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
338 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
243 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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0answers
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
3k views

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. ...
2
votes
1answer
36 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
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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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0answers
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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
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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
54 views

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
149 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
932 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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votes
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 ...
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0answers
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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
422 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.
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9answers
708 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.
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1answer
58 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
58 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
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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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votes
2answers
558 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
92 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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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 ...
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1answer
50 views

Discontinuities of an injective function from $\mathbb{R}$ to $\mathbb{R}$

It is well known that a monotonic function from $\mathbb{R}$ to $\mathbb{R}$ can have only countably many discontinuities. Question: Is it true that an injective function from $\mathbb{R}$ to ...
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0answers
44 views

Are there any other non-differentiable that came be constructed from summation besides the Weierstrass function?

So, I'd like to see conditions on a function $f(n,t)$ such that $F(t)$ from, $$F(t)=\sum_n f(n,t)$$ Is continuous over a non-zero range, but is nowhere differentiable. The range of the summation ...
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2answers
255 views

Are compact spaces characterized by “closed maps to Hausdorff spaces”?

It is well known that any continuous map from a compact space to a Hausdorff space must be a closed map. Does this fact characterize compactness? That is, if for a space $X$, every continuous map to ...
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0answers
168 views

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
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19answers
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Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
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2answers
180 views

Isometry of a metric space with proper subset

In Irving Kaplansky's "Set Theory and Metric Spaces", exercise 17 on page 71 asks for an example of a metric space which is isometric to a proper subset of itself. Any infinite discrete space and any ...
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3answers
590 views

Draw a non-planar graph whose complement is a non-planar graph

I have been teaching myself graph theory. I am stuck at solving this problem on my own. Please provide an example of such a graph. What approach would you take to draw such a graph?