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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2
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4answers
69 views

An example of a space $X$ which doesn't embed in $\mathbb{R}^n$ for any $n$?

Apologies if this has been asked before, but couldn't find it. The definition of embedding that I'm using is this: Suppose $X$ and $Y$ are topological spaces. We call a function $f:X\rightarrow ...
1
vote
1answer
67 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 ...
7
votes
1answer
131 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 ...
5
votes
2answers
231 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 ...
5
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1answer
54 views

McLain's characteristically simple group

Given a field $\mathbb{F}$, one can construct the free $\mathbb{F}$-module over the set of rational numbers. Assume that $(v_x)_{x \in \mathbb{Q}}$ is a basis of this linear space (indexed by the ...
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3answers
2k views

Need some help on a non-example of equicontinuity

In an attempt to better understand the definition of an equicontinuous family of continuous functions, I want to find a simple non-example. My intuition says that the family ...
171
votes
32answers
9k 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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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 ...
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2answers
95 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
votes
3answers
141 views

Examples of quasigroups with no identity elements

If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general). What would be some examples of quasigroups without an identity ...
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1answer
29 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 ...
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0answers
36 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
33 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
votes
1answer
34 views

Is this an example of a sequential non-Fréchet–Urysohn space?

Let $X$ be the set $X = \{ (0,0) \} \cup \{ (\frac{1}{n},0) : n \in \mathbb N \} \cup \{ (\frac{1}{i},\frac{1}{k}) : i,k \in \mathbb N \}$. Points of the form $(\frac{1}{i},\frac{1}{k})$ are ...
2
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1answer
96 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 ...
2
votes
0answers
47 views

specific magma examples

Give an example of a magma $S$ such that $S$ has a zero and $S$ has a left zero divisor that is not a right zero divisor an example of a magma with an identity such that there is an element with ...
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 ...
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1answer
51 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 ...
5
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3answers
1k views

Basic examples of monoids?

What are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply because they ...
42
votes
9answers
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
vote
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 ...
0
votes
1answer
94 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
37 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
72 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
2answers
108 views

Product of quotient maps and quotient space that is not Hausdorff

I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff.
0
votes
1answer
68 views

Normal subgroup question/example [duplicate]

Give an example(s) of a group $G$ and $H\leq G$ (i.e., $H$ is a subgroup of $G$) where $H$ is not normal in $G$. What about $S_3$ and $\langle(1\;2)\rangle=\{(1),(1\;2)\}$? [Since ...
5
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2answers
127 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 ...
1
vote
2answers
69 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 ...
2
votes
2answers
173 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 ...
0
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1answer
37 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 ...
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
89 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 ...
3
votes
1answer
52 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 $S^n(M)$? ...
4
votes
1answer
55 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 ...
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
31 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
1
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1answer
85 views

Extraneous solutions.

I just learned of extraneous solutions on the internet and thought, "could you both lose and gain solutions in one equation?" I think that, yes, you should be able to do that. However I haven't been ...
3
votes
2answers
74 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 ...
0
votes
1answer
38 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!
3
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0answers
74 views

understanding a theorem in c*algebra

I want to understand the following theorem: Theorem: Let A and B $C^*$-algebras with A unital, and let $\varphi:A\to B$ a bounded linear selfadjoint map such that: for every self-adjoint elements ...
1
vote
1answer
41 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
69 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 ...
7
votes
4answers
92 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.
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0answers
30 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.
29
votes
5answers
3k views

An example of an easy to understand undecidable problem

I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, ...
0
votes
1answer
28 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 ...
3
votes
1answer
31 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
51 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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2answers
1k views

What is the intuition between 1-cocycles (group cohomology)?

This is, I'm sure, an incredibly naive question, but: is there a simple explanation for why one should be interested in 1-cocycles? Let me explain a bit. Given an action of a group $G$ on another ...