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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2answers
38 views

does there exist nondifferentiable $f$ that is continuous AND has all directional derivatives $D_u$ AND $D_u=\nabla f\cdot u$?

There are many standard examples of functions $f:\mathbb{R}^2\to\mathbb{R}$ that possess all directional derivatives at a point and yet fail to differentiable or even continuous there. The most ...
2
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3answers
67 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 ...
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0answers
36 views

Draw a bad deformation of the unit circle

Dear counterexamplists, The goal is draw a continuous deformation of the unit circle that completely envelops the origin in a "worst way possible": It has to have the property that if you draw an ...
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1answer
42 views

How could I find a non-closed linear subspace $X$ of $l^{2}$ , such that $ l^{2} \ne X + X^{\perp}?$

How could I find a non-closed linear subspace $X$ of $l^{2}$ , such that $l^{2} \ne X + X^{\perp} ?$
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1answer
55 views

Example of decreasing sequence of sets with first set having infinite measure

I was wondering if someone could please give me an example of a sequence of decreasing sets where the first set has infinite Lebesgue measure; i.e., $\{B_{n}\}_{n=1}^{\infty}$ such that $m(B_{1}) = ...
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3answers
535 views

Connected, locally connected, path-connected but not locally path-connected subspace of the plane

I am looking for a set in the plane (with respect to the natural Euclidean topology) that is connected, locally connected, path-connected but not locally path-connected. I did not find one in ...
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1answer
25 views

Counter example: Sobolev Spaces

I need to show that the embedding of $H^1(\mathbb{R}^N)$ in $L^2(\mathbb{R}^N)$ is not compact. I need to find a sequence $(u_n)\subset H^1(\mathbb{R}^N)$ bounded such that there is no subsequence ...
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1answer
34 views

Is there any examples of a Banach algebra which every ideal of it, is a maximal ideal?

Is there any examples of a Banach algebra which every ideal of it, is maximal ideal? Or, Is there any conditions which turn all of the ideals of a Banach algebra to maximal ideals?
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1answer
10 views

$f\in C^{1}(0,1)$, $f(1/2)=0$ and $f>0$ in $(0,1)\setminus\{1/2\}$, then $\lim_{x\to1/2}f(x)/(1/2-x)^a=0$

I'm trying to prove the following claim (conjecture). Let $f\in C^{1}(0,1)$ be such that $f(\frac{1}{2})=0$ and $f>0$ in $(0,1)\setminus\{\frac{1}{2}\}$. Prove that $$ ...
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2answers
82 views

Convergence in $L^1$ norm, but not point-wise a.e.

As part of a course assignment, I'm asked to find a sequence of functions that converges in $L^1(\Omega \subset \mathbb{R})$, yet does not converge point-wise a.e. My thought is that such a sequence ...
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2answers
44 views

Why is the set of horizontal vector fields a vector space but not a Lie algebra?

I can't think of a concrete counter-example that shows a horizontal vector field is not necessarily a Lie algebra. Is there any easy example for this?
2
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1answer
38 views

An example of a map that is not unramified in a specific way

In Milne - Étale cohomology, the definition of an unramified morphism is given as follows (here I am using standard notation for sheaves, stalks, maximal ideals in the stalk and residue fields): Let ...
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1answer
45 views

An example of an unbounded uniformly continuous function on the open ball of $\ell_2$

It is a consequence of total boundedness of bounded intervals in $\mathbb{R}$ that uniformly continuous functions on such intervals are bounded. What is the best example of an unbounded uniformly ...
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2answers
60 views

Is the functions $f+g$ one to one?

I am reviewing for a test and there is this question. Functions $f: \mathbb R\to \mathbb R$, $g: \mathbb R\to\mathbb R$ are both one to one on the set of real numbers $\mathbb R$. Is the function ...
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7answers
1k views

An example showing that van der Waerden's theorem is not true for infinite arithmetic progressions

One of the possible formulations of Van der Waerden's theorem is the following: If $\mathbb N=A_1\cup \dots\cup A_k$ is a partition of the set $\mathbb N$, then one of the sets $A_1,\dots,A_k$ ...
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1answer
217 views

Find two discontinuous functions whose product is continuous

Find $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g: \mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is not continuous at 0, $g$ is not continuous at 3, and $f(x)g(x)$ is continuous everywhere. What ...
0
votes
1answer
147 views

Find a strictly increasing function $ f$ with $ f'(1)=0$

Find a strictly increasing function $ f$ with $ f'(1)=0$. I've found the function $f(x)=\frac{x^3}{3}−x^2+x$ But I don't know how to prove that the function is strictly increasing.
2
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2answers
68 views

Does a sub-$\sigma$-algebra of a countably generated $\sigma$-algebra have to be countably generated?

Assume that $A \subseteq B$ are $\sigma-$algebras and $B$ is a countably generated (separable) $\sigma-$algebra. Now my question: Is it possible that $A$ is not countably generated? I'm ...
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1answer
52 views

Looking for a counterexample in convergence of random variables

I was wondering is it possible that a sequence of $k$-dimensional random variables, $\{\mathbf{X_n}\}$ converges componentwise, but not jointly? What I mean is ...
0
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1answer
114 views

Sum of two odd functions is always odd.

The sum of two odd functions (a) is always an even function (b) is always an odd function (c) is sometimes odd and sometimes even (d) may be neither odd nor even The answer ...
0
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1answer
86 views

Vague convergence of absolutely continuous measures to discrete, or vice versa

Can a sequence of absolutely continuous probability measures converge vaguely to a discrete probability measure? Can a sequence of discrete probability measures converge vaguely to an absolutely ...
2
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0answers
79 views

Homology/cohomology for the uninitiated [closed]

I have heard of (co)homology occurring in many mathematical contexts and I vaguely suspect that it non-trivially relates different subjects. Also that it somehow relates to category/topos theory, ...
0
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0answers
34 views

If $S$ is closed in $\mathbb{R}^n,$ then $A$ is open in $S$ if and only if $A$ is open in $\mathbb{R}^n.$

Show, by counter-examples, that the following is not true: $(i)$ If $S$ is closed in $\mathbb{R}^n,$ then $A$ is open in $S$ if and only if $A$ is open in $\mathbb{R}^n.$ Let $S$ be the closed ...
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1answer
64 views

Counterexample to show that the interior of union may be larger than the union of interiors [duplicate]

I know the identity $\operatorname{int}(C \cup D) \supset \operatorname{int}(C) \cup \operatorname{int}(D) $. I need to find a counterexample showing that equality does not hold in general. Could ...
2
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1answer
42 views

Do uniformly grey sets of positive density exist?

Let us call a set $A\subset \mathbb{R}^2$ uniformly grey if the measure of its sections is constant, but not full. (There may be a standard name for this; I would be glad if someone tells me.) Formal ...
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2answers
151 views

Providing counterexamples to the claim: If $|A \cap B| < |A|$ then $|A|>|B|$

I'm having problems answering the following question: Give a counter example to show that the following statement is false: For any sets $A$ and $B$, if $|A \cap B| < |A|$ then ...
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votes
4answers
138 views

Can a relation be both symmetric and antisymmetric; or neither? [closed]

Can some relation be at the same time symmetric and antisymmetric? And, can a relation be neither one nor the other? Please give me an example for your answer.
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1answer
30 views

Show that $\forall x\varphi\vDash\varphi[t/x]$ may not hold if $t$ is bound for $x$ in $\varphi$.

Show that $\forall x\varphi\vDash\varphi[t/x]$ may not hold if $t$ is bound for $x$ in $\varphi$. Solution: Let $x=x_0$, $t=x_1$ and $\varphi=P(x_0,x_1)$. Then we have $\forall x_0(\exists ...
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1answer
191 views

Direct sum counterexample

Let $V$ be a vector space over a field $\mathbb{F}$, and let $W_i\ (i=1,...,4)$ be four distinct nonzero subspaces of $V$. Suppose that $W_i \cap W_j =$ {$0$} for all $i\ne\ j$. Is it true that the ...
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2answers
18 views

If I take pre-images of an increasing subset of the image, do their measures converge to that of the range?

Consider an arbitrary function $f:[0,1]\to\mathbb{R}_+$ and consider the sequence $$m^n=\mu(\{x\mid f(x)\leq n\}),$$ where $\mu$ is the Lebesgue measure. In essence, $m^n$ measures the pre-image of ...
0
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1answer
32 views

What is an example an injective holomorphic function whose derivative is zero at a point?

Let $G$ be open in $\mathbb{C}$. Let $f:G\rightarrow \mathbb{C}$ be an injective holomorphic function. Is it possible that at some point $z_0\in G$, $f'(z_0)=0$?
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0answers
68 views

Intuition for almost sure convergence = fast enough convergence in probability

I know the meaning of convergence in probability and almost convergence. From zero-one law, we can derive that if a sequence of random variables converges in probability fast enough, then it converges ...
2
votes
1answer
148 views

What does a Godel sentence actually look like?

My understanding is that Godel's first theorem says that there is a sentence that is in a sense true in a formal system F but cannot be derived in that system. I then hear that Godel actually goes on ...
0
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2answers
36 views

Counterexample to Absolute Continuity condition (for measures)

I'm taking an Analysis class and today we proved the following lemma: Let $\nu$ be a finite measure. Then $\nu\ll\mu$ iff for all $\varepsilon>0$ there exists $\delta>0$ s.t ...
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1answer
19 views

Best $g$ Approximation Poset maps

Let $P, Q$ be posets, $g: P\rightarrow Q$ be a monotone map. Let $x\in Q$. By a $g$-approximation of $x$ we mean an element $y\in P$ with $x\leq g(y)$. A best $g$-approximation of $x$ is a ...
0
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5answers
49 views

Proof of sets. Need an example

My question is to show that $X-(Y \cup Z)$ is a subset of $(X-Y) \cup (X-Z)$. I already did the proof for that and understand that but the second part is to give an example to show that in general, ...
8
votes
1answer
43 views

Can intersection of two manifolds be $xy=0$

I know that if two manifolds intersect transversally then their intersection is a manifold. But I was trying to construct an example where the intersection is not a manifold. But I still do not see ...
0
votes
1answer
64 views

Idempotent projection operators in a Hilbert space

Let $H$ be a Hilbert-space and $S$ be a sub(Hilbert)space such that: $$H = S \oplus S^\perp$$ Then the projection operators are defined as: $$P_S: H\to S; x = u + v \mapsto u \quad\quad P_{S\perp}: ...
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votes
1answer
108 views

If $x$ is an irrational number, then $3x^2 + 2$ is an irrational number? [closed]

Prove or disprove using direct or indirect proof that if $x$ is irrational, then $3x^2 + 2$ is irrational? Thank you in advance.
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2answers
33 views

Counting finite zeros among infinite zeros

Let $G$ be open in $\mathbb{C}$ and $f:G\rightarrow \mathbb{C}$ be an analytic function. Let's denote $deg(f,z)$ to mean the multiplicity of a zero $z$ of $f$, and $Z(f)$ to mean the set of zeros of ...
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2answers
103 views

Is there a sequence that has all real numbers as cluster points?

I recalled this interesting proposition from my real analysis course, for which the answer is true but I forgot the construction of such a sequence. I remember that's a point to get to the answer: ...
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2answers
39 views

Function examples [closed]

I have the following: Give example of sets F, G, R, S so that: F $\subseteq $RxS and G={(y,x): (x,y) $\in$ F) and : F and G are not Functions. F is a Function but G is not. F is not a Function but G ...
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1answer
468 views

Is there a function that doesn't have a derivative?

I was wondering if such a function exist. I'm comfortable with derivatives of polynomial functions, and some other basic functions, but I'm wondering if there could exist a very complicated function ...
1
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4answers
161 views

If squared series is convergent then series is convergent absolutely?

If $\sum_{i=0}^\infty a_n^2$ is convergent, does that imply $\sum_{i=0}^\infty a_n$ is convergent? How can I prove that? Thanks for the help.
0
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0answers
50 views

Examples of left-topological compact semigroups

I was reading chapter in Todorčević's book Topics in Topology (LNM 1652, DOI: 10.1007/BFb0096295) which deals with the semigroup $\beta\mathbb N$. Several results about left-topological compact ...
3
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1answer
51 views

Connected preimage of quotient map

I'm reading a paper on the fundamental groups of quotient spaces, and thought of the following question: Let $f: X \to Y$ be a quotient map with $X$ locally path connected and path connected, and ...
2
votes
1answer
46 views

Product of RREF versus RREF of product

Let $A$ and $B$ be two matrices of arbitrary shape where the number of columns of $A$ is the same as the number of rows of $B$. Is it always true that $$\textbf{rref}[A]\cdot \textbf{rref}[B] = ...
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1answer
100 views

Closed, simply connected manifolds which are not spheres

In 2 or 3 dimensions, every closed simply connected manifold is a sphere. In the smooth category, I suppose you could take exotic smooth structures to give examples of closed simply connected ...
2
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2answers
54 views

Need a counterexample related to Interior and Closure [duplicate]

Let "cl" denotes closure, "int" denotes interior.I'm looking for a Single example of a subset $A$ of some topological space $X$ where all the following sets are unequal: $1.$A $2.$ int(A), ...
6
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0answers
80 views

Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...