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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Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
13
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Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
12
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175 views

Hopf-like monoid in $(\Bbb{Set}, \times)$

I am looking for a nontrivial example of the following: Let a monoid $A$ be given with unit $e$, and two of its distinguished disjoint submonoids $B_1$ and $B_2$ (s.t. $B_1\cap B_2=\{e\}$), ...
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Does there exist a graph $G$ such that every edge is contained in a unique Hamiltonian circuit, that is not a cycle graph?

Suppose $G$ is an (undirected, simple) finite graph. If $G$ is a cycle graph, then each edge of $G$ belongs to a unique Hamiltonian circuit. Does there exist a non-cycle graph $G$ with this property?
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Exhibit a subset of $\mathbb A^2$ which is constructible, but not locally closed

A subset of a variety is locally closed if it is the intersection of a closed subset with an open subset; it is constructible if it is a finite union of locally closed subsets. Suppose that the base ...
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50 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
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168 views

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
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Request for counter examples in group theory

I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory). I am particularly interested in ...
5
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41 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
5
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62 views

What is an importance of Gaussian and Eisenstein rings?

Among quadratic integer rings, $\mathbb{Z}[i]$ and $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ have their special names, namely Gaussian integers and Eisenstein integers respectively. I guess this is named so ...
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127 views

Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
5
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42 views

A locally connected metric space of first category

I wonder if there exists a locally connected metric space of first category. I proved a theorem which assumes a space to be connected locally connected hereditarily Baire and metrizable. I ...
5
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92 views

Example of charts on $\mathbb{R}$ that are $\mathcal{C}^r$ compatible but not $\mathcal{C}^{r+1}$ compatible.

Is there a simple example of two charts where this is the case? I'm struggling to think up one.
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110 views

An interesting space inspired by Mrowka's space

The example is from Bell M G. First countable pseudocompactifications[J]. Topology and its Applications, 1985, 21(2): 159-166.. Let us recall some necessary definitions firstly: Let $X$ be a ...
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153 views

Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
5
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254 views

Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
4
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56 views

Simpler version of dogbone space construction

In "The cartesian product of a certain nonmanifold and a line is $E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold, $B$, such that $B\times \Bbb ...
4
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68 views

All possible total orderings of a finite set are isomorphic. What are some other examples of this phenomenon?

All possible total orderings of a finite set are isomorphic. I find these kinds of results remarkable. Here's a few more. Assume that $S$ is a finite set. Then: All possible field structures on $S$ ...
4
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39 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then ...
4
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66 views

An example of a nilpotent group

Is there an example of a nilpotent group such that $G/G'$ is (non-trivial) torsion-free while $G$ is not? I cannot think of any example of this kind and I think that it is not proved any result like ...
4
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about the closure of by pointwise convergence of sequences.

In the book "Integration Theory" (LNM315, K.Bichteler, p.65) a family $\mathcal{F}$ of real function is called full if it is close by pointwise convergence of dominated (by some element of ...
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129 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
4
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Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
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60 views

Self-duality in a lattice

Is there any finite self-dual lattice $(X,\le)$ such that there is not any self-duality $f:X\to X$ such that $f\circ f = 1_X$? Let $f,g:X\to X$ be a self-dualities. Then $f^{-1}\circ g$ is an ...
4
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90 views

A sequence of subsets of an infinite group.

Is there an infinite group $G$ such there is not any sequence $(A_n)$ of its subsets such that always $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?
4
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109 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
4
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87 views

Does there exist an infinite solvable group with no normal abelian subgroups?

This is impossible if $G$ is finite and solvable, because then $G$ has a (nontrivial) minimal normal subgroup $A$, which can be shown (using a trick) to be abelian. I'm trying to mimic the same ...
4
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103 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
4
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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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Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
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Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
4
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182 views

Relative Homology and Quotients

Are there any topological spaces $X$ with subspaces $A$ such that $H_n(X,A)$ is not isomorphic to $H_n(X/A)$? I've been trying some familiar spaces, but everything seems to be me an isomorphism ...
4
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244 views

Interesting applications of the cofinite topology?

Background: I'm doing some expository writing on intuitionistic logic and I have been toying with the idea of demonstrating its applicability via models where the denotations are taken from a Heyting ...
3
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67 views

Find two homeomorphic topological spaces and a bijective continuous map between them which is not homeomorphism.

I'm aware that it is duplicate, but I'd like to know whether my example is appropriate or not. Let our function $f$ be on the set $\mathbb{Q}\cap\mathbb{Z}$ induced by standard topology of a line. ...
3
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0answers
28 views

Monomorphisms into direct products

Let $G$ be a group. I am interested in the following property: For any groups $A,B$ and monomorphism $G \hookrightarrow A \times B$, either $G \hookrightarrow A$ or $G \hookrightarrow B$. For ...
3
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74 views

mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
3
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Separability of conjugacy classes in conjugacy separable semidirect products.

We say that group $G$ is conjugacy separable if for every $g \in G$ the set $g^G = \{cgc^{-1} \mid c \in G\}$ is closed in the profinite topology on $G$, i.e. for every $f \in G \setminus g^G$ there ...
3
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49 views

Find an example such that $\tau$ is a stopping time and $\mathcal{F}_\tau$ and $\mathcal{F}_\infty$ differ on $\{\tau = \infty\}$.

I need to find an example such that the following is true: $\tau$ is a stopping time and $\mathcal{F}$ is a filtration defined on $\mathbb{R}_+$. Let $\mathcal{F}_\tau$ denote the stopped ...
3
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156 views

Why this is not a Banach space

When reading about functional analysis I encountered the following example of a Banach space: $ C^1 ([0,1])$ endowed with the norm $\|f\| = \|f\|_\infty + \|f'\|_\infty$. where $\|\cdot\|_\infty$ ...
3
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56 views

Left continuous magmas with no fixed points

Let $X$ be a compact Hausdorff topological space, and $*: X^2\rightarrow X$ an associative map (so that $(X, *)$ is a semigroup) which is left continuous (for all $s\in X$, the map $t\mapsto ts$ is ...
3
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203 views

An open problem on general topology

There is an open problem in this paper: J. van Mill, V.V. Tkachuk, R.G. Wilson, "Classes defined by stars and neighbourhood assignments", Topology and its Applications, Vol. 154, Issue 10, 2007, pp. ...
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Moscow space-Examples

A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G‎_{δ}$‎‎‎ ‎-subsets of $X$ . For example, Every first countable $T_1$ ...
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Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
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91 views

Azumaya algebra and its subalgebras

I remind you that an Azumaya algebra $A$ is a central and separable algebra. Now, I know that if $A$ is an algebra over a skew-field or over a local ring then there exists a subalgebra $S$ of $A$ such ...
2
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34 views

What is an illustrative example of a Finsler manifold?

I've attempted to get a bead on Finsler manifolds before attending an upcoming seminar that involves them. I've done some reading in the literature and think I understand that they are similar to ...
2
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25 views

A modified Shapiro's Tauberian theorem? Proof or counterexample

Let $\{a(n)\}$ be a nonnegative sequence such that $$\sum_{n\leq x}a(n)[x/n^{2}]=x^{2}\log x + O(x^{2})$$ for all $x\geq 1$, where $[y]$ denotes the greatest integer $\leq y$. Is true that the ...
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54 views

Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
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Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
2
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73 views

Finding a weight function with a specific property

I am looking for a (smooth, quickly decaying) function $w : [0,\infty) \rightarrow \mathbb{R}$ such that $$w(t) \cdot \int_{0}^{t} \frac{1}{w(2x)} dx $$ is absolutely integrable on $[0,\infty)$. ...
2
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On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...