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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WHat is an example of a countable subset of $[0,1]$ whose Jordan content is 1 and Lebesgue measure is 0?

E.Stein Real analysis p.41 Exhibit a countable subset $E\subset [0,1]$ such that $J(E)=1$ while $m*(E)=0$. Here, $m*$ denotes the outer Lebesgue measure and $J$ denotes the Jordan content. ...
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46 views

Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
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Distributive lattices (interpretation of distributivity)

Simple counter example Ok, there is a very simple counter-example ^^ : This lattice isn't distributive, because $M=x\wedge(a\vee b)=x>0=(x\wedge a)\vee(x\wedge b)=m$, but for all $n<N$ the ...
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2answers
167 views

Matrix multiplication question (diagonal matrices)

Suppose $AB = BA$ and $A^2+B^2 = I$, where A and B are complex matrices. My feeling is that this implies that both A and B are diagonal matrices. But I'm having trouble proving it. Appreciate any ...
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1answer
35 views

Smallest Graph that is Regular but not Vertex-Transitive?

I'm trying to find the smallest graph that is regular but not vertex-transitive, where by smallest I mean "least number of vertices", and if two graphs have the same number of vertices, then the ...
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1answer
15 views

Function differentiable at one point and nowhere else continuous.

Is it possible to construct such a function? Just wondering. Specifically, I am thinking of $f:\mathbb{R}\to\mathbb{R}$ such that $f'(0)$ exists and $f$ is discontinuous for all ...
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1answer
33 views

A characterization for subgroups.

Let $G$ be a group and $a_0,a_1,...,a_n\in G$ and $$A=\{a_0,a_1,...,a_n\}$$ and $$(\forall m\le n)(\forall i\le m)(a_{i}a_{m-i}\in A)$$ Is $A$ a subgroup of $G$? How if $G$ is abelian?
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Blowup of ODEs in the presence of local Lipschitzianity?

Pardon me if the question is trivial, but I am failing to decide it. Assume that we are given an ODE system $\dot{x} = f(x)$ with positive initial conditions $x(0)$ and know that $f$ is locally ...
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2answers
43 views

Give an example of a non-separable subspace of a separable space

I'm trying to find an example of a non-separable subspace of a separable space. What kind of examples are there?
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2answers
62 views

A topological space which is Frechet but not Strictly-Frechet.

Let $X$ be a topological space and $q \in X$. $X$ is strictly Frechet at $q$, if, for all $A_n \subset X, q \in \bigcap_{n \in \omega} \overline {A_n}$ implies the existence of a sequence $q_n \in ...
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1answer
52 views

What would be an example such that $aH=bH$ but $Ha \neq Hb$?

Let $H$ be a subgroup of a group $G$. Assume $aH=bH$ for some $a,b\in G$ What would be an example such that $Hb\neq Ha$? I cannot imagine what would be.. Moreover, if $H$ is infinite, ...
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35 views

Smash products of pointed spaces is really not associative

The canonical bijective map $\mathbb{N} \wedge (\mathbb{Q} \wedge \mathbb{Q}) \to (\mathbb{N} \wedge \mathbb{Q}) \wedge \mathbb{Q}$ is not an isomorphism of pointed spaces (i.e. homeomorphism), see ...
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1answer
39 views

Counterexample for Hartogs' Extension Theorem

I'll refer to Hartogs' Extension Theorem as it is stated in Wikipedia (https://en.wikipedia.org/wiki/Hartogs%27_extension_theorem#Formal_statement). I am trying to find a counterexample to show that ...
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3answers
79 views

Give an example of non-normal subspace of a normal space.

We know that a closed subspace of normal space is normal. My question was: why should other subspaces not work and then i came up with a counterexample. It is peculiar that any subspace of regular ...
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1answer
40 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
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1answer
29 views

Example of a Riemann integrable function which is not a simple function

I'm looking for an example of a Riemann integrable function which isn't simple? I know that all simple functions $f: I \rightarrow E $ ( where $I \subset \mathbb{R}$ is an interval and $E$ is a ...
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1answer
66 views

Example of two finitely isomorphic structures (or $\omega$-isomorphic) that are not partially isomorphic?

One can define several maps between given structures in order to portray similarities or differences. The concept of isomorphism displays a deep structural overlapping, while for instance an ...
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1answer
19 views

Sperners lemma how to mark internal vertices

Was reading sperners lemma from this http://www.math.hmc.edu/funfacts/ffiles/20001.4.shtml Couldn't understand certain things How to mark internal vertices? I could have mark some other number for ...
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19 views

Non-subdifferentiable convex function

Is there any convex function $f$ on a norm space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$? Thanks in advance.
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113 views

Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
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33 views

Reference request: An example of a false conjecture with a very large number as the first counter example

I recall that there was some conjecture, something that I believe involved prime numbers, and was believed to be true (as it was checked up to a relatively large number) until a counter example was ...
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2answers
23 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
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1answer
53 views

Violation of the irrelevant alternative criterion of fairness in a pairwise comparison

I am teaching my students about the fairness criteria for voting system, working up towards arrow's impossibility theorem. One of the voting methods is called the pairwise comparison method: voters ...
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1answer
60 views

Inner Product on a Vector Space over a field besides $\mathbb R$ or $\mathbb C$?

Are there any fields with vector spaces you can define an inner product over besides subfields of $\mathbb C$? I know that you'd want the field to contain an ordered subfield, so it must have ...
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1answer
35 views

Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
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1answer
107 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
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2answers
151 views

What is $\sum_{n=0}^{\infty}|a_nz^n|^2=\frac{1}{2 \pi}\int_{-\pi}^{\pi}|f(ze^{it})|^2dt$ for?

Let $f : \mathbb C\rightarrow \mathbb C$ be an analytic function : $f(z)= \sum a_n z^n$ It holds that $$a_n z^n= \frac{1}{2 \pi}\int_{-\pi}^{\pi}f(ze^{it})e^{-int}dt$$ and ...
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2answers
54 views

Peculiar examples to the Stone Representation Theorem

The Stone Representation theorem states that every Boolean algebra is isomorphic to a field of sets. That is, a Boolean algebra whose elements are sets, and sums, products, negation are union, ...
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888 views

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Example of two open balls such that the one with the smaller radius contains the one with the larger radius. I cannot find a metric space in which this is true. Looking for hints in the right ...
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1answer
14 views

A non-closed $p$-group.

Are there a Hausdorff topological group $(G,\mathcal T)$ and and a non-closed $p$-group $P\le G$ ? a $p$-group where $p$ is a prime number, is a group $P$ such that $$(\forall a\in P)(\exists n\in ...
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2answers
51 views

Sigma-algebra requirement 3, closed under countable unions.

The requirement for sigma-algebra is that. It contains the empty set. If A is in the sigma-algebra, then the complement of A is there. 3. It is closed under countable unions. My question relates ...
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1answer
52 views

A question about polynomials in $K[x_1,x_2,…,x_n]$ and there permutations

Let $K$ be a field. Let $n$ be a positive integer and $P$ be a non-symmetric polynomial in $K[x_1,x_2,...,x_n]$. $S_n$ acts on $K[x_1,...,x_n]$ in an obvious way. Let $P_1,P_2,...,P_r $ be the ...
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1answer
51 views

Does $X=[0,\omega_1]$ satisfy $S_1(\Omega,\Omega)$?

Definition: An $\omega$-cover of a topological space $X$, is an open cover $\mathcal U$, such that, for any finite set $C \subset X$, there exists an open set $U \in \mathcal U$, such that, $C \subset ...
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1answer
24 views

Square of absolute value of a function different than square of function

How come if f is measurable, we might have $|f|^2\neq f^2$? Can you provide an example? I think it is true if f is real.
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1answer
29 views

Counterexample of Separation Theorem in topological vector space

The Separation Theorem states that: If $A$ and $B$ be two disjoint convex subsets in a vector space $X$ and one of them has nonempty core (algebraic interior) then there exists a linear functional ...
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1answer
27 views

Closed, bounded and convex subset in $X^*$ but not $w^*$ closed.

Banach-Alaoglu states that: If $X$ is a topological vector space then the polar of any neighborhood of the origin is $\sigma(X^*,X)$ compact. Especially, if $X$ is a norm linear space then the closed ...
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1answer
45 views

Groups of order $p^aq^br^c$ containing elements of order $pq$, $qr$, and $pr$, but not $pqr$

Let $G$ be a solvable group of order $p^aq^br^c$ (for distinct prime $p,q,r$) containing elements of orders $pq$, $qr$, and $pr$, but no element of order $pqr$. Furthermore, assume that $G$ is ...
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1answer
21 views

Convex function that has a finite limit at infinity

Can someone give me an example for a convex function that has a finit limit at infinity ?
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1answer
21 views

The space with countable complement topology (example 20 in “Counterexamples in topology”)

As a continuation to this question: Given the space of countable complement topology on $X$, where $X$ is an uncountable set. (example 20 in "Counterexamples in topology"). We know that $X$ is not ...
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1answer
37 views

Indiscrete rational extension for $\mathbb R$ (examp 66 in “Counterexamples in topology”)

As a continuation to this question: Let $X$ be the Indiscrete rational extension for $\mathbb R$ (examp 66 page 88 in "Counterexamples in topology"). Let $\langle \mathcal{U}_n: n \in \mathbb{N} ...
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0answers
18 views

An example of a Lindelöf topological space which is not $\sigma$-compact

I am looking for an example of a Lindelöf topological space which is not $\sigma$-compact. I have looked in Counterexamples in Topology, but, if I am not wrong, all the examples there which meet my ...
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Discrete HW Question

Show that For all sets A and B, $A^{C} \cup B^{C} \subset (A \cup B)^{C}$ is false by a counterexample.
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intersection of locally compact Hausdorff topologies.

Are there locally compact Hausdorff topologies $\mathcal T, \mathcal S$ on a set $X$, such that $\mathcal T\cap \mathcal S$ is a Hausdorff but not locally compact topology on $X$?
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Condition to separability of a Banach space.

I am trying to prove the following statement: Let X be a Banach space and $X^{*}$ its topological dual space. If there exists a countable family of functions $(f_{n})_{n} \subset X^{*}$ such that ...
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82 views

An example of a noncommutative PID

It's well known that when a ring $R$ is a PID, every submodule of a free $R$-module is free. I'm interested in cases when the converse holds -- that is, in rings $R$ which have the property that every ...
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1answer
48 views

Counterexample of separation theorem

I'm trying to know a counterexample for separation theorem: If $A$ and $B$ are two disjoint convex set in a topological vector space $X$, one of them has nonempty interior, then there exists $f\in ...
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35 views

Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
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1answer
67 views

Hard to find counterexample for $\partial (\partial A) = \partial A$

In an exercise I've proven that $\partial(\partial A) \subset \partial A$, for any $A\subset X$, where $X$ is a topological space and $\partial$ in this case stands for the boundary. Apparently, in ...
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1answer
56 views

A question on Fourier Transform

Is there a function which is not absolutely integrable but which has a continuous fourier transform? I know that if a function is absolutely integrable then the fourier transform is continuous but I ...
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1answer
114 views

Is there a nonempty open bounded subset of plane whose boundary contains no 1 dimensional interval?

Someone asked a question here which hasn't received a correct answer because everyone seems to be misinterpreting the question. I would like to ask the question again. Does there exist a nonempty ...