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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12
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163 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 ...
10
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259 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 ...
8
votes
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144 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
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45 views

Is a topological space determined by its components and their quotient?

Given connected topological spaces $X_i$ and a totally disconnected space $Y$, is there a unique topological space $X$ with components homeomorphic to $X_i$ and $X/\sim$ homeomorphic to $Y$? ($\sim$ ...
6
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115 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
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78 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
votes
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107 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 ...
5
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78 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
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98 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
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134 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
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238 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
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184 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
42 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
votes
0answers
56 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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0answers
56 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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0answers
26 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 ...
4
votes
0answers
125 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
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77 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
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67 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
224 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
83 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 $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?
3
votes
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94 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathcal{Top}$, $k$-closed subset $Y\subset X$, means $u^{-1}(Y)$ is closed in $C$ for any $u: ...
3
votes
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65 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 ...
3
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137 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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53 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
188 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
142 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
79 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
123 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 ...
3
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88 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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31 views

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 ...
2
votes
0answers
33 views

Natural examples of multiplicatively-graded rings

A ring $A$ is multiplicatively graded if, as an additive group, it decomposes as $$A=\bigoplus_{n\geq 1} A_n$$ and the product satisfies $A_n \times A_m \rightarrow A_{nm}$. A natural example of ...
2
votes
0answers
30 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 ...
2
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0answers
37 views

A real-valued separately continuous function discontinuous everywhere

I need an example of a real-valued separately continuous function that is discontinuous at each point. This is either a well-known folklore fact or a burning problem in separate versus joint ...
2
votes
0answers
31 views

an example that property $\delta$ does not imply property $\gamma$

In this article, two properties are mentioned at page 153: property $\gamma$: If $\mathcal U$ is an open $\omega$-cover of $X$, then there is a sequence $G_n \in \mathcal U$, with $\underline {Lim} ...
2
votes
0answers
51 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
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41 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
30 views

Kullback-Leibler $KL(p,q)\neq KL(q,p)$

I'm doing a course of Artificial Intelligence and in my homework I must to provide a counter example to show that the Kullback-Leibler distance is not a symmetric function of its arguments: $$ ...
2
votes
0answers
26 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
48 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
94 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
80 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
51 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
104 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
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0answers
54 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
46 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
120 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
121 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
105 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 ...