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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107 views

Can I Have Some More Examples Of Uniform Spaces Please?

I have been reading about uniform spaces and topological groups. There does not look to be a lot of literature on the topic, much less accesible literature, and the books that I have been reading do ...
2
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1answer
55 views

Basic Multilinear regression question for finding examples or counterexamples.

Hello Wise mathematicians! I have few quenstions about Multi linear regresstion. I've been asked from my friend, but I have very weak knowledge background from that field. It seems my friend is in ...
3
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1answer
116 views

If $f(x)$ is positive and decreasing, can $xf(x)$ have more than one maxima?

Assume that $f(x)$ is positive and decreasing on $[0,1]$ with $f(0)=1$ and $f(1)=0$. We see that $xf(x)$ is 0 at 0 and 1 and is positive in between so it must have a maxima. Is it possible that ...
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569 views

Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1 $} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
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1answer
123 views

Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff ...
12
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393 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 ...
3
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3answers
91 views

Counterexample for a complex analysis proof

I'm having troubles coming up with a counterexample for the following: If $|f(z)|$ is continuous at $z_0$, then the function $f(z)$ is continuous at $z_0$ for complex numbers. I know I need a $f(z)$ ...
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1answer
46 views

Moduli space of stable principal $G$-bundles on a compact Riemann surface.

Let $C$ be a compact Riemann surface. I'm looking for some references in order to try to understand what is the moduli space of stable principal $G$-bundles on $C$, where $G$ is a simple Lie group. ...
2
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1answer
370 views

Examples of affine schemes

I have to introduce affine schemes as topological spaces in a small seminar. Could you suggest me three examples of affine schemes in order to put in evidence the correlation among traditional ...
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0answers
94 views

Is this integral a counter example to this theorem?

I may have misunderstood the proposition, but I thought it was: Let $f$ be a function $[a,b]\times I\to \Bbb R$, where $I$ is some real interval. Then a sufficient condition for ...
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2answers
108 views

Can we find a subset of $Spec(R)$ not quasi-compact?

If $R$ is a commutative ring with unit, we can easy prove that $Spec(R)$ is quasi-compact. However can you give me an example of $R$ such that a subset $A \subset Spec(R)$ isn't quasi-compact?
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116 views

Are there simple examples of Banach spaces with no non-trivial Clifford Isometries?

By a Banach space $X$ I mean, a complete normed vector space and by a Clifford isometry I mean a surjective isometry $\gamma$ of $X$ such that the distance $d(\gamma x, x)$ is constant on $X$. ...
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1answer
88 views

Compact spaces where not all compact subsets are closed

A topological space $(X,\tau)$ is called $C-C$ iff the closed sets in $X$ coincide with the compact sets in $X$. A topological space is called a $US$-space provided that each convergent sequence has ...
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1answer
285 views

Example of rings of the same positive characteristic that do not embed into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got me wondering: Give an example of commutative rings $A$ and $B$ with $\operatorname{char}A=\operatorname{char}B$ such that ...
3
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3answers
128 views

Can an object be universal with respect to several properties?

Does anyone know an example of an object $A$ in some category $\mathcal C$ such that $A$ satisfies at the same time more than one universal property? Of course, when I say that $A$ is universal with ...
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0answers
64 views

Are there motivating examples for graphs with negative edge weights?

Every example I come up with for a directed graph with negative edge weights seems contrived in some way. Can anyone name some real ones? Even better if they aren't obviously networks to start with.
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3answers
410 views

A $T_1$ space which is not Hausdorff

More precisely, assume the following definitions. Definitions. Let $S$ be a topological space. $S$ is a $T_1$ space if, whenever $s_1 \neq s_2$ there exists an open set $U_1$ such that ...
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1answer
93 views

Is there a counter example?

Is this true? If $f$ and $g $ are continuous functions , and if $f\circ g$ is closed(open) , neither $g$ nor $f$ is necessarily closed(open)?
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1answer
106 views

Counterexample for the logical argument

Give a counter example to show the argument is not valid by using formulas from a particular structure interpreting the language. $\forall x\exists y (r_1xy)$ $\forall y \exists x (r_2xy)$ then, ...
2
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2answers
613 views

How to prove if something is false or true?

I'm a bit stuck at a task i'm working on here. Here is the task: Show statement "If $(P \rightarrow Q)$ and $(Q \rightarrow R)$ is true, then $(P \rightarrow R)$ is true by a counter ...
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1answer
72 views

Example of a ring $R$ and ideals $A$ and $B$ of $R$ such that $AB\neq A\cap B$.

Give an example ofa ring $R$ and ideals $A$ and $B$ of $R$ such that $AB\neq A\cap B$. I know that $AB\subset A\cap B$. But I can't find an example such that $AB\neq A\cap B$.
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451 views

The tree property for non-weakly compact $\kappa$

In my previous question, Weakly-compact cardinals, I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2_2$. One of which was that ...
0
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1answer
67 views

Are there many compact, zero-dimensional, first-countable spaces with $d(Z)=\mathfrak{c}$?

Are there many compact, zero-dimensional, first-countable spaces with $d(Z)=\mathfrak{c}$, where $d(\cdot)$ stands for density (i.e. least size of a dense subspace)? Thanks for your help!
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1answer
276 views

what's the role of fiber bundles play in understanding the base space?

Suppose $\pi: E\to B$ be a fiber bundle, and assume $B$ is connected, I want to know: 1, What do the fiber bundles (e.g., the covering spaces) on $B$ help understand the topology, or the structure on ...
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2answers
246 views

Is there a first countable, 0-dimensonal, locally compact, lindelof, non-compact space?

Is there a first countable, 0-dimensonal, locally compact, lindelof, non-compact space in which all non-empty open sets have $\pi$-weight $\mathfrak c$? It also can be seen here. Thanks for your ...
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3answers
155 views

abstract algebra example book

It's very exciting when you can use the theory to solve "lower level" problems. For example, I'm looking forward to understanding why the quintic equation is not solvable. In the undergraduate ...
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0answers
80 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 ...
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1answer
142 views

Counterexample to Eisenstein criterion

We know Eisenstein criterion about irreducibility of polynomials: if $q(x) = x^n + a_{n-1}x^{n-1} + \dots +a_0 \in \mathbb{Z}[x]$ is such that $\exists p$ prime number with $ p \mid a_{i} \ \forall i ...
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2answers
1k views

how to solve a third degree equation of complex roots and coefficients

It's not a homework it came in one of our exams and I didn't find anything on the internet that is a high-school level. please give me any hint or answer to solve this in a noncomplicated way. solve ...
4
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1answer
89 views

A pseudocompact space with $G_\delta$-point

Is evert $T_2$ pseudocompact space with $G_\delta$-points always first countable? Does there exist a counterexample? Thanks ahead.
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Intuitive meaning of Limit Supremum?

I am trying to understand the difference between the following two equations: $$\bar{P} = \limsup_{t \to \infty}\frac{1}{t} \sum_{\tau = 0}^{t-1}E\{P[\tau]\} < \infty$$ and $$\bar{P} = \lim_{t ...
4
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4answers
159 views

Why Does the existence of $\frac{\partial f}{\partial x}$ not imply that $\frac{\partial f}{\partial x}$ is continuous?

For $f(x)$, the existence of $f'(x)$ implies the continuity of $f(x)$. And I am assuming that it also implies the continuity of $f'(x)$. My question is why in a function $g(x,y)$, is the existence ...
3
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2answers
149 views

Groups with $\wedge$-irreducible trivial subgroup

Suppose $G$ is a group satisfying the following condition: $$H \cap K = \{1\} \implies H = \{1\} \;\text{ or }\; K=\{1\}$$ for any two subgroups $H$, $K$, i.e. the trivial subgroup is ...
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1answer
38 views

A question on locally compact metalindelöf spaces

Is there a locally compact metalindelöf space $X$ with $|X|> 2^{\mathcal c}$, where $\mathcal c=2^\omega$? Thanks for helps.
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4answers
851 views

Can we get un-obvious results by defining sophisticated topologies?

What I originally found so interesting about general topology was how general a type of thing a topology is, and how the terminology open, closed, compact, continuous, convergence et cetera means ...
4
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1answer
66 views

For this periodic continuous $g:\Bbb R\to \Bbb R$, and $f_n(x):=g(x/n)$, does $\{f_n\}_{n=1}^\infty$ converge uniformly?

I can not find a counterexample although I have the feeling it is not true. Let $\ g: \mathbb{ R} \rightarrow \mathbb{R}$ continuous function $ \forall x \in \mathbb{R} \ g(x+1) = g(x)$ $g(0) = 0$ ...
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2answers
82 views

A sequence $(t_{k,n})$ st $\forall n$, $\sum_{k=1}^n t_{k,n}=1$ but $\lim_n t_{k,n} \neq 0$.

I'm looking for an example of a sequence $(t_{k,n})$ such that for all $n$, $$\sum_{k=1}^n t_{k,n}=1$$ but $$\exists k: \ \lim_{n \to \infty}t_{k,n} \neq 0.$$ I've been looking at Toeplitz' lemma for ...
2
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1answer
82 views

Example of an integrable function with an entire extension and whose derivative only vanishes at infinity

I am looking for a function $f : \mathbb{C} \to \mathbb{C}$ with the following properties: $f$ is entire. $\int_{-\infty}^\infty |f(t)| \ dt < \infty$ i.e. the restriction of $f$ to the real line ...
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1answer
44 views

The product of metrizable space and one point lindelofication of the discrete space

Let $X$ be the one point lindefication of a discret space of cardinality $\omega_1$ and $Y$ is any metrizable space. Is $X \times Y$ always Lindelöf? If I may ask more, does a metric space $Y$ with ...
6
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2answers
293 views

Does There exist a continuous bijection $\mathbb{Q}\to \mathbb{Q}\times \mathbb{Q}$?

Does there exist a continuous bijection $\mathbb{Q}\to \mathbb{Q}\times \mathbb{Q}$? I am not able to find out how to proceed.
4
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2answers
187 views

Normal products of groups with maximal nilpotency class

Let $H$ be a nilpotent group of class $a$ and $K$ a nilpotent group of class $b$. If $H$ and $K$ are normal subgroups of a group $G$, then we know that $HK$ is a normal nilpotent subgroup of $G$ and ...
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0answers
90 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 ...
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1answer
81 views

How $|\mathbb R|$ is not weakly compact

$\mathbb R$, of cardinality $|\mathbb R|=2^{\aleph_0}$ is not weakly compact. So there is a function $f$ from $[{\mathbb R}]^2$ (the subsets of $\mathbb R$ of size 2) to $\{0,1\}$ such that there is ...
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1answer
43 views

seek a special matrix

I am seeking the following example, maybe it is easy to construct, but I have no idea now besides aimless computation. Could anyone give me a matrix $A\in SL_n(\mathbb{Z})$ for some $n$, such that ...
34
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5answers
2k views

False beliefs about Lebesgue measure on $\mathbb{R}$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
5
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1answer
72 views

Counterexample to a lemma about modules

Let R be a ring with identity and not necessarily commutative. Let $M_1, M_2$ be left $R$-modules with submodules $S_1, S_2$ respectively such that $M_1/S_1 \cong M_2$ and $M_2/S_2 \cong M_1.$ Is it ...
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2answers
484 views

Is there a discontinuous function on the plane having partial derivatives of all orders?

If one requires simply the existence of partial derivatives of first order rather than all orders, then a standard example is the function $$ f(x,y) = \left\{\begin{array}{l l} \frac{2xy}{x^2+y^2} ...
6
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2answers
789 views

Free modules over commutative rings. [duplicate]

Free modules over a commutative ring $R$ with $1$ have well-defined rank. I have been wondering if there is a ring $R$ such that there are free modules $M'\subset M$ with ...
3
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3answers
226 views

How common is it for a densely-defined linear functional to be closed?

I've always held the vague belief that any densely-defined operator encountered "in nature", if it isn't bounded, is probably at least closable. But, today I noticed the following thing: Consider ...
2
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1answer
459 views

Counter examples in group theory [duplicate]

Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements 1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$ 2) If $G/N_{1}\cong ...