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)

10
votes
0answers
209 views

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 ...
9
votes
0answers
125 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
7
votes
0answers
136 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. $B_1\cap B_2=\{e\}$), ...
6
votes
0answers
81 views

Example where Čech and derived functor cohomologies don't agree.

I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces. Is there a simple example of a topological space ...
6
votes
0answers
112 views

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 ...
5
votes
0answers
95 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 ...
5
votes
0answers
67 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.
5
votes
0answers
96 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 ...
5
votes
0answers
126 views

Primes of the form $\dfrac{n^2-n+4}{2}$ 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
votes
0answers
233 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 ...
5
votes
0answers
155 views

A counter example in obstruction theory

Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map ...
4
votes
0answers
66 views

Fibrations with isomorphic fibers, but not Zariski locally trivial

I am looking for examples of fibrations $f:X\to Y$ where the fibers are all isomorphic, but $f$ is not Zariski locally trivial. In particular, I am interested in understanding how much such examples ...
4
votes
0answers
108 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
4
votes
0answers
75 views

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 $\{H_\alpha: \alpha < \kappa\}$ ...
4
votes
0answers
62 views

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
votes
0answers
215 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
votes
0answers
125 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
votes
0answers
44 views

Lindelöf property and $\omega$-covers

Let $X$ be a Lindelöf topological space. Does this imply that every $\omega$-cover has a countable subcover which is also an $\omega$-cover? if not, is there an example of a topological Lindelöf space ...
3
votes
0answers
177 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. ...
3
votes
0answers
130 views

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 $T_1$ ...
3
votes
0answers
76 views

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], ...
3
votes
0answers
87 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
votes
0answers
16 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 ...
2
votes
0answers
47 views

intersection of locally compact Hausdorff topologies.

Are there locally compact Hausdorff topologies $\mathcal T, \mathcal S$ on a set $X$, such that $\mathcal T\cap \mathcal S$ is a Hausdorff but not locally compact topology on $X$?
2
votes
0answers
29 views

Specific examples of “hard-to-factorize” numbers

I am playing with different factorization algorithms such as Pollard-Rho and Pollard-(p-1) or Quadratic Sieve. I would like to test these algorithms in some examples. I know I can multiply two large ...
2
votes
0answers
44 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 ...
2
votes
0answers
25 views

A question on the classical Mrowka space

Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset ...
2
votes
0answers
46 views

Restricted continuity implies continuity

When teaching calculus, we instruct students to calculate multivariate limits using the following theorem: If $\gamma$ is a smooth curve with $\gamma(0) = a \in \newcommand{\R}{\mathbb{R}}\R^n$ ...
2
votes
0answers
87 views

Is there a “natural” norm in the space of all sequences?

Let $X$ be $\mathbb{R}^\mathbb{N}$, the space of all real-valued sequences. Is there a natural norm in this space?
2
votes
0answers
77 views

supple, non flabby sheaf

Can anyone give an example of a sheaf that is supple, but not flabby? Consider sheafs $\mathcal{F}$ of Abelian groups over $X$. it is flabby if for any $U$ open subset of $X$, the restriction ...
2
votes
0answers
50 views

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. ...
2
votes
0answers
96 views

Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
2
votes
0answers
53 views

Inequality change in $\mathbb{E}[ \max |\cdot|] $ due to $\max$

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W)=1$. Find $m$, a locally-bounded function $f:\mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow ...
2
votes
0answers
44 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 ...
2
votes
0answers
103 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 ...
2
votes
0answers
115 views

Is inversion sequentially continuous in SOT?

Let $A_n \overset{SOT}{\to} A$ where $A$ is invertible. Does $A_n^{-1} \overset{SOT}{\to} A^{-1}$? Does $A_n^{-1} \overset{WOT}{\to} A^{-1}$? EDIT: Forgot to mention $\{A,A_n\}\in\mathscr{B(H)}$ ...
2
votes
0answers
114 views

Examples of reductive groups of dimension $4$ and semisimple rank $1$

This is the problem: Exhibit three reductive groups of dimension $4$ and semisimple rank $1$ which are pairwise nonisomorphic (as algebraic groups). I know that for any reductive group $G$ of ...
2
votes
0answers
49 views

examples of 'continuous bases of functions,' like the Fourier transform

For suitable choice of a one-parameter family of functions $\{ g_w:\mathbb{R} \rightarrow \mathbb{C} \}_{w\in \mathbb{R}}$, the following two statements are equivalent (modulo sets of measure $0$): ...
2
votes
0answers
101 views

List of large classes of functors providing morphisms

I recently posted this question on mathoverflow and it was closed as being too localized. I am hoping to more precisely say what I mean here. I recently learned, through my Topology coursework, that ...
1
vote
0answers
34 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 ...
1
vote
0answers
45 views

Heineken and Mohamed Groups

I was trying to understand the costruction of the Heineken and Mohamed groups. (II) Every proper subgroup of $G$ is subnormal and nilpotent. Lemma 1. If $G$ is a non-nilpotent soluble group ...
1
vote
0answers
28 views

nearest point and closed complement of a subspace in norm spaces

It is well-known that in any Hilbert space $H$, each closed subspace $Y$ admits a closed complement $Y^\perp$. This result also implies that there exists a best approximation point to $Y$ for any ...
1
vote
0answers
66 views

Nice convergent subsequence of $\cos(n)$.

This question is related to a few questions which have been posted on the website : Is there a limit of $\cos(n!)$ Converging subsequence on a circle The limit of $\sin(n!)$ Because of the ...
1
vote
0answers
25 views

smooth in one direction, nowhere differentiable in the other

Can anyone think of an example $u\in L^2(\mathbb{R}^2)$ such that $u$ is smooth in $x_1$ but nowhere differentiable in $x_2$?
1
vote
0answers
16 views

Complete lattice mod a congruence relation.

Suppose $(X,\le)$ is a lattice and $R$ is a congruence relation on it. If $X$ complete, is $\frac{X}{R}$ complete?
1
vote
0answers
40 views

Resolvable spaces

a space $X$ is called a resolvable space if it is expressible as a union of two disjoint dense subsets. I want to find a resolvable but not lindelof space? Is there any example such a space?
1
vote
0answers
18 views

Is a extension of a premeasure preserves outer-measure generated by the premeasure?

I have proved the follow: Let $X$ be a set. Let $S$ be a semi-ring of subsets of $X$. Let $\mu$ be a premeasure on $S$. Let $\overline{\mu}$ be a premeasure on a ring generated by ...
1
vote
0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
1
vote
0answers
32 views

Another question on spaces with calibre-$\aleph_1$

Let $X$ be a strongly monotonically monolithic space with calibre-$\aleph_1$. Must $X$ be Lindelof? I know $e(X)=l(X)$ for a strongly monotonically monolithic space. So to prove that $X$ is ...
1
vote
0answers
63 views

How can I calculate $\mathrm{Spec}(\mathbb{Z}_{(3)})$? And $\mathrm{Spec}(\mathbb{Z}_3)$?

Let $\mathbb{Z}_{(3)}$ be the localization (in $\mathbb{Z}$) of the ideal generated by $3$. So I have to put in $\mathbb{Z}$ all the inverses of the complement of $(3)$. How can I calculate ...