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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15 views

smooth function whose (n+1)th derivative is defined only on a propersubset of the domain of the nth, and the radius contract to 0

So I'm basically wondering if there exists such a function, whose (n+1)th derivative is defined only on a proper subset of the domain where the nth derivative is defined, and with the property that ...
5
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2answers
60 views

Is every finite weak group a group

Definition: Let $W$ be a set and $\circ:W\times W\rightarrow W$ be a function. We say that $(W,\circ)$ is a a weak group iff there exists unique $e\in W$ such that $\forall x\in W[x\circ e=e\circ ...
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1answer
84 views

Caught in the net

I'm reading through some notes one locally convex spaces ("lcs" from now on) analysis and there the following version of the Banach-Steinhaus theorem is given Theorem (Banach-Steinhaus) $\quad$ ...
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1answer
26 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
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2answers
56 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
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2answers
530 views

Is every group the automorphism group of a group?

Suppose $G$ is a group. Does there always exist a group $H$, such that $\operatorname{Aut}(H)=G$, i. e. such that $G$ is the automorphism group of $H$? EDIT: It has been pointed out that the answer ...
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5answers
4k views

List applications of sets & relations in science/business/tech that a highschooler can understand

What are some applications of sets & relations in science/business/tech that a highschooler can understand? To kindle a young mind, what examples can be given?
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2answers
477 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
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0answers
39 views

Examples of PIDs and prime ideals

(a) Give a specific example of a PID with exactly two prime ideals. Give a brief proof of your answer. (b) Give an specific example of a PID with infinitely many prime ideals. Give a brief proof of ...
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1answer
41 views

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 ...
1
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1answer
63 views

Proofs about continuity and convergence in topological spaces

I'm working on the following exercise: Let $f:(X,T)\to(Y,S)$ and $x\in X$. Prove that if $f$ is continuous at $x$ then if a sequence $\{x_n\}$ converge to $x$ we have $f(\{x_n\})\to f(x)$, show ...
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4answers
60 views

I have a conjecture on local max/min , can any of you propose a contradiction?

If $f$ is a non-piecewise function defined continuous on an interval $I$, and within that interval $I$, there exists a value $x$, such that $f`(x)$ (derivative of $f$) does not exist , then at that ...
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1answer
90 views

Changes in the hypotheses of a mean-value theorem

For $X \subset \mathbb{R}^d$ open, we define $$ C^1(X) := \left \{ f : X \to \mathbb{C} : f \text{ is a function s.t. } \frac{\partial f}{\partial x_j} \text{ exists and is continuous for } j = 1, 2, ...
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1answer
110 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this?? I guess $A$ must be an $G_{\delta}$ set which is dense in ...
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6answers
382 views

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? [duplicate]

I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers ...
3
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2answers
72 views

A real continuous periodic function with two incommensurate periods is constant.

I think I have a proof for the statement, but I can't think of a counter-example when $f: \mathbb{R} \to \mathbb{R}$ is not continous. Here's the problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a ...
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2answers
51 views

Is a bounded continuous function defined on $\Bbb R$ differentiable?

Is a bounded continuous function defined on $\Bbb R$ differentiable? Why so? The query is fueled by the following question: Let $f : \Bbb R \rightarrow \Bbb R$ be a bounded continuous function. ...
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1answer
38 views

Are there finite “similar” non-isomorphic groups

Let $G_1,G_2$ be two groups.We say that $G_1,G_2$ are similar iff for every integers $a_1,a_2,...,a_n\in \{1,-1\}$ and every function $f:\{1,...,n\}\rightarrow\{1,...,n\}$ we have the following: ...
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0answers
46 views

maximum modulus principle analytic function

I am trying to show that: Let $f$ be analytic on a given closed unit disc $D$ then prove that for every $k\in\mathbb N$ there is $w\in Bd(D)$ such that $|f(z)-w^{-k}|≥1.$ where z is in the unit disc ...
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2answers
71 views

Category In Which Not All Free Objects Exists

I am trying to think of a category in which not all free objects exists. I thought this might be the case in sets (I thought I might be able to violate the uniqueness ) but I couldn't get anywhere so ...
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3answers
228 views

Does Gödel's Completeness Theorem still hold even if the set of variables is finite?

Let $L$ be a first order language with a finite set of variables. Let $T$ be a consistent set of formulas of $L$. Does it follow that there exists a model for $T$?
146
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29answers
8k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
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0answers
68 views

Maximal monotone operator without convex domain?

I'm looking for an example of a (multi-valued) maximal monotone operator $A$ mapping a Banach space $X$ into its dual $X^*$ such that the domain $D(A)=\{x\in X: Ax\neq\emptyset\}$ is not convex. ...
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1answer
41 views

Non-isomorpic uncountable dense linear orderings (a textbook example)

We know that two countable dense open orderings are isomorphic, and we know that this is generally not true for uncountable structures. Now I quote from the textbook, that is "Model-Theoretic Logics": ...
4
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1answer
127 views

On $T_2$, first countable, countably compact space

As we know, For every $T_2$, first countable, compact space, its cardinality is not more than $2^\omega$. (See chapter 3 of Engelking's book.) However, I want to know whether the result is ...
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2answers
268 views

Epimorphism from GL(2,Z) to GL(2,Z)

Is there an epimorphism $f\colon \mathrm{GL}(2,\mathbb{Z})\to \mathrm{GL}(2,\mathbb{Z})$ which is not injective? Here, $\mathrm{GL}(2,\mathbb{Z})$ is the group of invertible $2\times 2$ matrices with ...
5
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1answer
196 views

non-residually finite group

Let $G$ be the subgroup of $\text{Bij}(\mathbb{Z})$ generated by $\sigma : n \mapsto n+1$ and $\tau$ which switches $0$ and $1$. How can we prove that $G$ is not residually finite? Is it hopfian?
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0answers
27 views

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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3answers
57 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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0answers
28 views

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 ...
3
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2answers
222 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 ...
2
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1answer
41 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
22 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 ...
2
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1answer
60 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
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1answer
36 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?
5
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1answer
134 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 ...
1
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2answers
52 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?
3
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2answers
63 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 ...
0
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1answer
66 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, ...
3
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0answers
65 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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3answers
100 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
57 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 ...
2
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9answers
6k views

Example of functions that are onto but not one-to-one

I have been preparing for my exam tomorrow and I just can't think of a function that is onto but not one-to-one. I know an absolute function isn't one-to-one or onto. And an example of a one-to-one ...
3
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1answer
57 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 ...
1
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1answer
86 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 ...
6
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2answers
139 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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1answer
60 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
24 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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0answers
34 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.
4
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1answer
67 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 ...