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 ...

learn more… | top users | synonyms (2)

1
vote
1answer
55 views

Show that every nearly compact space is almost compact space but the converse is not true

I am learning about the almost compact space and nearly compact space. I know that every nearly compact space is almost compact space but the converse is not true in general. So i need an example of ...
10
votes
1answer
165 views

What are some interesting counterexamples given by finite topological spaces?

According to Wikipedia, 'finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures.' I have been studying the book '...
0
votes
1answer
28 views

Counter example of Algebraic sets

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
3
votes
1answer
64 views

Does equal cardinality and one set being a subset of the other prove equality? [closed]

I'm currently solving a quite specific problem and in the final step I made a statement that can be generalised such that: $$((|A|=|B|)\wedge(A\subset B)) \implies (A=B)$$ Whilst this is clearly ...
1
vote
2answers
92 views

Which groups $G$ has the property that for all subgroups $H$ , there is a surjective map from $G$ to $H$?

I tried many examples , but i can't find any counterexample . But I guess there are many counter examples , and specific sorts of groups or subgroups have this property (e.g abelian groups or normal ...
2
votes
2answers
45 views

A function not differentiable at a point but whose derivative has a limit

Does there exist a function $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is continuous on a neighborhood of $0$, $f$ is differentiable at all $x$ close to $0$ except at $0$ itself, and $\lim_{x\to 0} f'...
1
vote
2answers
51 views

$\nabla ^2 G$ meaning

If $G$ is a function with three components. $G$ is actually a Green's function in my case, like $G( \textbf{x} , \xi)$ with $\textbf{x} = (x,y)$ and $\xi = (\xi _x, \xi _y)$ so I am guessing it is ...
1
vote
4answers
50 views

How can I provide a counterexample for this predicate logic problem?

I'm honesty still unsure of what a counterexample even is, and what I've found on isn't helping me much in the way of understanding. I'm hoping to get pointed in a correct direction. Predicates L(...
0
votes
0answers
22 views

Second derivative of a set in non-Fréchet space

In Fréchet (T1) topological spaces it's easy to prove that $A''\subseteq A'$, but the proof doesn't work without this assumption. What are some illuminating counterexamples when the space is not ...
1
vote
1answer
85 views

Counter example to Stone Weierstrass Theorem

If we miss some conditions of Stone Weierstrass Theorem, will this theorem still hold? I have come up with counter examples when we do not have compact metric space. But what if the function algebra ...
1
vote
0answers
36 views

Examples of Algebraic quantum groups

I am reading articles about algebraic quantum groups, which are defined (see A. Van Dael) as a regular multiplier Hopf algebra $(A,\Delta)$ for which there exists a non-zero functional $\varphi$ on $A$...
1
vote
0answers
53 views

Topological manifold but not “smooth”

I would like to know if there are examples of topological manifolds that not admit a smooth atlas, or more generally a differentiable structure of class $C^k$ for $k\geq1$. In the book "Introduction ...
1
vote
1answer
45 views

Can we have a continuous choice in the mean value theorem

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable. Must there exist a continuous function $g:\{(a,b)\in \mathbb{R}^2: a<b\}\rightarrow \mathbb{R}$ such that: For every two distinct real ...
9
votes
3answers
291 views

Counterexample: Continuous, but not uniformly continuous functions do not preserve Cauchy Sequences

I want to prove this: There exists a continuous function $f:\mathbb{Q}\to\mathbb{Q}$, but not uniformly continuous, and a Cauchy sequence $\{x_n\}_{n\in\mathbb{N}}$ of rational numbers such that $\{f(...
0
votes
0answers
25 views

Example of a non semisimple Lie Algebra over $\mathbb{C} $ such that $ L' = L$ [duplicate]

I wanted to show that if $L$ is a finite dimension semisimple Lie Algebra over $\mathbb{C}$, then $L' = L$ ($L' = [L,L]$) but the converse is not true. I could prove the statement but am unable ...
0
votes
1answer
40 views

Finding an example of an extension of length 2

I am working on extensions in general but for sake of simplicity we can assume it's a module here. I am interested in an extension of the form $$0\to B \to E_2\to E_1 \to A \to 0$$ which is an ...
2
votes
3answers
72 views

How to construct a sequence that has a subsequence convergent to every $k\in \Bbb{N}$?

How to construct a sequence $\{a_{n}\}^{\infty}_{n=1}$, such that for every $k\in \Bbb{N}$, $\{a_{n}\}^{\infty}_{n=1}$ has a subsequence convergent to $k$? A subsequence is such as $2,4,6,...$ in $1,...
0
votes
1answer
26 views

Equidecomposable examples

Decomposable: A set $S \subset \mathbb{R}^n$ is decomposable in $m$ sets $A_1,…,A_m \subset \mathbb{R}^n$ if there exist isometries $\phi_1,…,\phi_m:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that: ...
0
votes
1answer
55 views

Difference between flow and solution of ODE

I am reading Wikipedia's entry on Flow and it is not clear the distinction between solution of an ODE and the flow of an ODE. In particular it is clearly written $φ(x_0,t) = x(t)$, then what is the ...
2
votes
1answer
29 views

What would be a counterexample for a point which is a limit point of isolated singularities?

Let $G$ be open in $\mathbb{C}$ and $f:G\rightarrow \mathbb{C}$ be a function. Define $D$ as the set of points in $G$ at which $f$ is complex-differentiable. That is, $p\in D$ iff $\lim_{z\to p} (f(z)...
1
vote
1answer
42 views

Give an example for if … [duplicate]

Prove that if $H$ is a normal subgroup of a group $G$ and has index $n$, then for any $g\in G$ we have $g^n\in H$. Give an example for if $H$ is not normal, the mentioned statement is not correct. (...
1
vote
2answers
20 views

substitutional interpretation of quantifiers: examples?

About the differences between propositional logic and (first order) predicate logic, given that if my basis is propositional logic I have to substitute the universal and existential quantifiers with ...
0
votes
2answers
51 views

Propositional logic vs predicate logic: examples?

About the difference between the propositional logic and the (first order) predicate logic-> can you give me one or more remarkable examples which underly the differences and the similarities ...
4
votes
2answers
176 views

Are there powerful ways to use the topological definition of continuity in real analysis?

In the lectures for introductory real analysis, my professor repeatedly told the class that the topological definition of continuity (preimage of open is open) is the most powerful version of ...
4
votes
2answers
64 views

Group Operations/ Group Actions

I'm currently taking my first abstract algebra course and am learning about group actions, orbits, and stabilizers. I'm reading the Artin textbook and I am not very clear of what exactly a group ...
0
votes
1answer
32 views

Definition of a recursive ordinal

I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows: "...an ordinal $\alpha$ is said to be recursive if there is a recursive well-...
4
votes
4answers
113 views

Theorems Implying their Own Generalization [closed]

Are there any examples of theorems which were later found to imply their own generalization? Here's an example of what I mean: Hypothetically, suppose you proved Fermat's Little Theorem: $a^p \equiv ...
2
votes
4answers
125 views

Prove or disprove: If $n^3$ is odd then $n$ is odd.

If $n^3$ is odd, then $n$ is odd. I need to prove or disprove by means of counterexample why this is true or false. $\forall x P(x) = x^3$, $x = 1,3,5,7,9$ I am having a very difficult time ...
2
votes
1answer
40 views

Can we construct a continuous but nowhere differentiable surface?

In my analysis course we learned about the Weierstrass function which is continuous but nowhere differentiable, is it possible to make a surface which is continuous and nowhere differentiable?
1
vote
2answers
26 views

An example of a prime quotien ideal where his corresponding ideal is not prime

If $R$ is a comutative ring with identity ring and $K$ is an ideal from it, let $R'=R/K$ and $I$ an ideal of $R$ satisfy $K\subseteq I$ and $I'$ is the coresponding ideal of $R'$ (we knew that ...
4
votes
3answers
69 views

Trivial power of line bundle

I'm trying to understand the following: Thinking of a line bundle as a bunch of locally generating sections together with transition functions (which in this case are just multiplication by local ...
3
votes
3answers
76 views

Is there a countably compact sequential non-$T_2$ space that is not sequentially compact?

Let $X$ be a topological space. Definitions: $X$ is countably compact if every countable open cover of $X$ has a finite subcover or equivalently, every sequence in $X$ has a cluster point. $X$ is ...
0
votes
1answer
62 views

Example of graph with specific $\chi (G)$, $\omega (G)$, $\beta (G)$

Find an original example of a graph whose chromatic number does not equal its clique number, yet whose clique partition number equals its independence number. Chromatic number: $\chi(G)$ is the ...
6
votes
5answers
548 views

Example of a ring with an infinite inclusion chain of ideals [closed]

I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)
3
votes
1answer
125 views

Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either: Proofs i.e. pointwise equicontinuity is uniform ...
1
vote
0answers
53 views

Greedy algorithm fails to give chromatic number

Produce a graph and degree sequence for which the greedy algorithm fails to give the chromatic number. My first example is below- The first labeling uses 2 colors which is the chromatic number and ...
0
votes
1answer
49 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
1
vote
1answer
37 views

An example of smooth compactly supported function with non-vanishing Fourier transform

I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the ...
0
votes
1answer
41 views

Example of maximum modulus principle

As it's known , an holomorphic($\neq constant$) function $f:G\subseteq\mathbb{C}\rightarrow \mathbb{C}$ has maximum modulus on $\partial G$ . I wuold an example of a function holomorphic on a disk ...
5
votes
5answers
136 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
5
votes
5answers
550 views

A Compact Hausdorff Space with no Manifold Structure? [closed]

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
6
votes
0answers
33 views

What are types of coalgebras that are more naturally described by cooperads?

Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of endomorphisms and ...
3
votes
3answers
104 views

What exactly is the maximal solution of an ODE and why do we care?

I am reading these notes on the definition of a maximal solution of an ODE i.e. http://www.math.lmu.de/~philip/publications/lectureNotes/ODE.pdf But the definition is so abstract and no example is ...
4
votes
0answers
62 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
1
vote
1answer
40 views

An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and $\text{diam}(...
3
votes
3answers
88 views

Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it? Though there are ...
3
votes
3answers
82 views

Example of Topological Group

I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology. ...
0
votes
0answers
16 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
0
votes
0answers
28 views

Prove or Disprove If $T : R^n \rightarrow\ell_2$ is linear, then $T(A)$ is totally bounded in $\ell_2$ when $A ⊆ R^n$ is bounded

Prove or Disprove, If $ T: \mathbb R^n \rightarrow \ell_2 $ is linear, then It preserves total boundedness $ T(A) $ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded. I think ...
0
votes
1answer
36 views

problems to understand a special definition of “free graded commutative algebra” from lecture

I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., ...