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 [on hold]

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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0answers
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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
53 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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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
59 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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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 ...
2
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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
3
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1answer
23 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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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
44 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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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 ...
2
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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
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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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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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 $$ g(x) = \min_{y\in J} f(x,y) $$ ...
3
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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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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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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
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
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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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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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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0answers
12 views

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
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 ...
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
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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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
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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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 ...
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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
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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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1answer
51 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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1answer
84 views

What is a counterexample that a composition of covering maps not a covering map?

Let $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ be covering maps. What would be an example that $q\circ p:X\rightarrow Z$ is not a covering map? I saw a counterexample here, but it was too complex. Is ...
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2answers
50 views

Example of a non-polynomial function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x)$ is negative for $x<0$ and positive for $x \ge 0$.

I have a bunch of polynomial functions example easily (e.g. $x^2$), but have trouble coming up with a non-polynomial function. I was thinking of defining $f(x) = e^{-x}$ for $x<0$ and $f(x) = ...
2
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2answers
33 views

Double complex with exact rows

Let $(K^{p,q},\delta,d)$ be a double complex of modules. We assume that $\delta$ of degree $(1,0)$, $d$ has degree $(0,1)$ and $d$ and $\delta$ commute. Since $d$ and $\delta$ commute, then $\delta$ ...
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1answer
40 views

Deleting a contractible subspace is the same as deleting a point

Let $X$ be a topological space and $A$ and $B$ are subspaces of $X$. Suppose that $A$ is contractible. I know that taking the quotient does not affect the homotopy type, that is $X/A$ is homotopy ...
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1answer
96 views

$X$ is connected, $A \subset X$ connected, and $C$ a component of $X\backslash A$. Is $\overline A \cap \overline C \ne \emptyset$?

I'm trying to prove or disprove the following statement: If $X$ is connected, $A \subset X$ is connected, and $C$ a component of $X\backslash A$ then $\overline A \cap \overline C \ne \emptyset$. I ...
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1answer
60 views

Continuous and bounded imply uniform continuity?

I am thinking about this since couple hour. Is a continuous and bounded function $f:\mathbb R\to\mathbb R$ uniform continuous too? I didn't found a counter example and thus I tried to prove this like ...
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1answer
72 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
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1answer
35 views

Can this statement about winding number generalized?

Definition Let $\alpha$ be a path in $\mathbb{C}\setminus\{z_0\}$. Since $\mathbb{C}\rightarrow \mathbb{C}\setminus \{z_0\}:z\mapsto e^z$ is a covering map, $\alpha$ can be decomposed as ...
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0answers
20 views

What is an example that Euclidean function varies by a unit?

Let $R$ be a Euclidean domain and $f$ be a Euclidean function on $R$. (Not necessarily submultiplicative) Let $a,b$ be nonzero elements of $R$ such that $a=ub$ where $u$ is a unit of $R$. Is there ...
5
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1answer
61 views

Example of a surjective local homeomorphism that is not a covering? [duplicate]

Let $X$ and $Y$ be connected, locally path connected, and Hausdorff topological spaces. Can someone give me an example of a surjective local homeomorphism that is not a covering? I don't think this ...
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2answers
31 views

Continuous marginal distributions do not imply continuous joint distribution

I already proved the other implication. I need to find an explicit example that shows that if there is some random vector $(X,Y)$ and $X$ and $Y$ have both continuous marginal distributions, then ...
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2answers
43 views

Existence of an metric or a topology so that every subset is compact

Let $X$ be a infinite set. Is there a metric on $X$ such that every sub set of $X$ is compact? What about a topology on $X$? I think that if we can answer first question then we can answer the ...
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0answers
50 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
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1answer
74 views

Sobolev functions counterexample

Let $A=(0,1)^{d}$.Does anyone have a simple example of a funtion in $H_0^1(A)\cap H^2(A)$ that is not in $H^2_0(A)$? Thanks a lot.