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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7
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1answer
140 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
1
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1answer
20 views

Counterexample for continuous function over product topology without compactness

Suppose $f$ $(X,d_x)$: $\rightarrow$ $(Y,d_y)$ is a function between metric spaces, and $X \times Y$ has the product topology. The graph $G_f$ is the subspace $G_f$ = {$(x,f(x))$ | x $\in$ $X$}. If ...
1
vote
1answer
69 views

closed, convex, absorbing subset of a banach space

There is a nice theorem that every closed, convex, absorbing subset of a banach space includes an open ball arround $0$. Can you give an example where the theorem fails if we do not assume the subset ...
1
vote
1answer
30 views

What kind of functions lie in $L^2(\Omega )-H^{1/2}(\Omega )$?

It's well known that a discontinuous function does not lie in $H^{1/2}(\Omega )$ if $\Omega\subset\mathbb{R}$ is one dimension. My question is, For $\Omega\subset \mathbb{R}^3$, are there any simple ...
12
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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 ...
2
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0answers
38 views

Not continuous function with closed graph

I would like to see an explicit example of a function $f:R\rightarrow R$ which has a closed graph in $R^2$ but is discontinuous at every point in the real line.
1
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1answer
35 views

Closed Subsets of the Real Line that are Uncountable

If a subset of the real line is uncountable and closed, does it have to contain a closed interval? Is there any theorem related to this?
2
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0answers
62 views

Counter-example to exponential law for locally compact [non-Hausdorff] spaces

There is a natural bijection $$ \operatorname{Map}(X\times Y,Z)\cong\operatorname{Map}(X,\operatorname{Map}(Y,Z)),\quad f\mapsto(x\mapsto(f(x,{-})). $$ If $X$, $Y$, $Z$ are topological spaces one can ...
2
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1answer
105 views

Example of $\deg(fg)<\deg(f)+\deg(g)$

Let $R$ be an integral domain and $f,g\in R[X_1,...,X_n]$ where $n>1$. What is an example of a pair $f,g$ such that $\deg(fg)<\deg(f)+\deg(g)$? Moreover, i have proven that the units of ...
0
votes
1answer
24 views

Oscillating essential discontinuities exist?

Let $f$ be a function $\mathbb R \to \mathbb R$. According to Wikipedia an discontinuity of $f$ is essential if and only if either the left or the right limit is infinite or does not exist. Is it ...
1
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1answer
57 views

Continuity and interior

I have questions about the relation between continuity and interior based on the article ;Continuity and Closure At first I guess that there will be a property like $f:X\rightarrow Y$ is continuous ...
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0answers
37 views

Example of a function such that iterated integrals are equal

Is there any example of a function $f(x,y):[0,1]$x$[0,1]\to \mathbb R$ so that $\int_{0}^1\int_{0}^1f(x,y)dydx$ and $\int_{0}^1\int_{0}^1f(x,y)dxdy$ exists and are equal but $\int\int f(x,y)dydx$ does ...
1
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1answer
112 views

Examples for 1d finite element method

I am looking for some examples for my Finite Elements project (in one dimension). I have written code in MATLAB and would like to show a few examples of it working. I have one or two general examples ...
1
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2answers
41 views

Non-Hausdorff space such that all connected components are singletons

Is there a topological space $(X,\tau)$ such that $(X,\tau)$ is not Hausdorff; if $S\subseteq X$ and $S$ contains more than 1 point, then $S$ is not connected (with the subspace topology).
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0answers
39 views

Counterexample to exactness of functor from group representations to fixed points

I recently asked this question. Now, the answer there claimed that the functor $()^G:Rep_G\to Vect_{\mathbb{C}}$, where $Rep_G$ are complex representations of a group $G$, and $V^G=\{v\in V: ...
2
votes
1answer
66 views

What would be an example of free module such that cardinality of linearly independent set is greater than the rank?

Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has the IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=\mathrm{rank}(M)$. Let ...
5
votes
2answers
238 views

Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...
0
votes
1answer
38 views

Example of a subgroup that is not normal (not involving permutations)

It would be great if someone could give me an example of a group such that the following happens a $\equiv$ b (mod N) and c $\equiv$ d (mod N) but ac $\not\equiv$ bd (mod N) where N is a ...
2
votes
2answers
222 views

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable. I know how to find the eigenvalues, and diagonalizing ...
2
votes
1answer
75 views

Associativity of concatenation of closed curves from $I$ to some topological spaces $X$

I'm looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$. I tried to use $X$ like the Sierpinski space, but I can't find such closed curve.
0
votes
0answers
11 views

infinite intersection of jordan measurable sets

Is the infinite intersection of jordan measurable sets also jordan measurable? I´ve been trying to find a counterexample, but nothing so far. So is the statement true?
3
votes
1answer
69 views

Is the symmetric algebra direct sum of $k$-th symmetric powers?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $S(M)$ be the symmetric algebra of $M$ and $S^n(M)$ be $n$-th symmetric power of $M$. Then is $S(M)$ the internal direct sum of $S^n(M)$? ...
3
votes
1answer
90 views

Two categories sharing the same objects and morphisms

Is there a natural example of two categories $\mathcal{C}$, $\mathcal{C}'$ which have the same class of objects and the same class of morphisms, including source and target maps, but different ...
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2answers
70 views

Identity about a Functor

I'm moving my first steps in CT and suddenly after reading about functors, this question came up in my mind: Let $F \colon A \to B$ a functor between two categories $A$ and $B$, is it true that for ...
47
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10answers
1k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
1
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0answers
62 views

Examples for when differentiability fails

Let $l^1(\mathbb{N};\mathbb{R})$ be the set of all sequences $\mathbb{N}\to\mathbb{R}$ such that $\sum_{n\in\mathbb{N}}|x_n|<\infty$ for all $x\in l^1(\mathbb{N};\mathbb{R})$, together with the ...
1
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3answers
33 views

Sets with one accumulation point

Are there any more examples of sets in $\mathbb R$ that have one accumulation point apart from convergent sequences? I can´t think of any
0
votes
1answer
41 views

Uniform convergence in series definitions of functions

Are there examples of well-known functions which are defined as the limit of a sequence of functions (for example, power series definitions) and are not uniformly convergent? Thanks!
1
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1answer
51 views

Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.

My question is: do you know any examples when $X$ and $Y$ are both normally distributed, but the two dimensional vector $(X,Y)$ is not? I found some example in book, but I don't understand it. The ...
3
votes
1answer
71 views

What is an example of $R\otimes_R M$ not isomorphic to $M$?

Let $R$ be a ring and $f:R\rightarrow R$ be a ring homomorphism. ($f(1)=1$) Let $R$ be given the left operation as the ring operation on $R$ and the right operation as $x•r=xf(r)$, so that $R$ is an ...
3
votes
2answers
81 views

Which optimization problem to use when I want to explain the concept of optimization to a layman?

I am looking some problem, which would be: Easy to understand Hard to solve intuitively Touch our everyday lives I am doing research in optimization using evolutionary computation. When people ...
7
votes
4answers
93 views

Example for $\dfrac{p_1p_2-1}{p_1+p_2}$ being odd natural number .

If $p_1,p_2$ are odd prime numbers , is it possible that $\dfrac{p_1p_2-1}{p_1+p_2}$ is odd natural number greater than 1.
0
votes
1answer
29 views

Integral on set sequence is not convergent to their countable intersection

Let $(\Omega,S,\mu)$ be a measure space. Suppose $f$ is integrable on $A_1\supset A_2 \supset A_3\dots$, a decreasing sequence of measurable sets $\{A_n\}_n\subset S$ and denote ...
5
votes
2answers
139 views

Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale

I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
3
votes
1answer
38 views

Examples for almost-semirings without absorbing zero

What is an instructive example of a set $X$ equipped with two monoid structures $(X,+,0)$, $(X,\cdot,1)$, such that $+$ is commutative, the distributive laws hold, but $0 \cdot x = 0$ or $x \cdot 0 = ...
2
votes
0answers
56 views

A metric space of which the geodesic is not a metric

The text book in my course has an exercise about finding a metric space whose (usual) length metric is not a metric. It wants me to find a metric space $(X,d)$ satisfying $d'(x,y)=0 \ \ $for some ...
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0answers
93 views

P = NP, NP example problems in our daily life

For a little presentation for school, i want to try to explain the P=NP? Problem. I'm searching for examples for daily life NP-problems. (example: is making the weather forecast a NP problem?) And if ...
1
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0answers
34 views

Similar function to Sinc function?

I am interested in a function which at $x=0$ starts from approximately $1$ and as you go on it decreases periodically to $0$ in a similar fashion to the Sinc function.
0
votes
1answer
39 views

Infinite sum of random variables is infinite

I am trying to better understand this statement and the assumptions made: If $X_1,X_2,\ldots$ are non-negative independently and identically distributed random variables with $P(X_i>0)>0$, ...
3
votes
1answer
59 views

Example of equivalence relation where reflexivity/symmetry is not trivial

For basically any equivalence relation that I've encountered in my mathematical studies, the only problematic part (if at all) was proving transitivity. Is there a "real-world" (i.e. appearing in some ...
14
votes
6answers
845 views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
0
votes
1answer
95 views

Practical examples/implementation details for Gauss-Seidel method

I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory , I would like to have some practical examples for this method (a system of linear equations with n>=1000, ...
1
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2answers
138 views

Examples of Separable Spaces that are not Second-Countable

In this post I give an example of a separable space that is not second-countable. What are other good examples?
2
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1answer
49 views

Is there a particularly simple example of geometric descent?

I'm looking for a particularly simple and familiar example of descent in geometry or topology in order to motivate the general definition. I'm not counting the definition of the arrow category ...
4
votes
1answer
57 views

Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...
6
votes
2answers
129 views

If $\frac{a+b}2$ is rational, can we say that $a,b$ are rational?

The question is if it's given that $$ {a+b\over 2} \in \Bbb Q $$ prove or disprove $a,b \in \Bbb Q$. Since it is to disprove, I tried the following method by using examples. Take $$a = 1 + \sqrt{2} ...
1
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1answer
14 views

What is an example of a module over a division ring with two different ranks?

Let $R$ be a division ring and $M$ be an $R$-module. What is an example of $M$ and two bases $A,B$ of $M$ such that $|A|\neq |B|$?
3
votes
3answers
256 views

Examples of interesting / non-trivial manifolds that are direct products

What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds. Say, an ...
0
votes
2answers
42 views

Is a hausdorff perfect space which is not first countable neccessarily uncountable?

Counterexamples in Topology has a couple of countable spaces which aren't first countable, but none of them are perfect spaces. I'm looking either for a theorem that says that such a space can't exist ...
1
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1answer
79 views

Counterexample in optional stopping martingale

Problem: Give an example of submartingale $\{X_n\}$ with $\sup_nE |X_{n-1}-X_n|<\infty$ and stopping time $N$ with $E[N]<\infty$ such that $\{X_{n\wedge N}\}$ is not uniformly integrable. ...