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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55 views

Let $G_1,G_2$ be groups with 2 subgroups respectively $H_1,H_2$ satisfying certain conditions, must $|G_1:H_1|=|G_2:H_2|$

Let $G_1,G_2$ be groups with two subgroups respectively $H_1,H_2$ such that there is a bijection $f:G_1\rightarrow G_2$ and $f|H_1$ is a bijection between $H_1,H_2$. Must $|G_1:H_1|=|G_2:H_2|$ ? ...
4
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1answer
174 views

Counterexample to second derivative test when f''(x) is not continuously differentiable

When looking over true/false questions on previous midterms, one of my conscientious students said: "If f is defined on an open interval containing c, f'(c)=0, and f''(c)>0, then c is a local min of ...
6
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1answer
251 views

Existence of minimal non-zero prime ideals: Counter examples?

Let $R$ be an integral commutative ring with unit. If $R$ is noetherian, then every ideal has finite height, in particular, there exist minimal non-zero prime ideals if (and only if) $R$ is not a ...
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1answer
280 views

Example to $\lim f(x)g(x)$ may not exist

Let $A\subset\mathbb{R}$, $c$ a cluster point of $A$ and $f,g:A\rightarrow \mathbb{R}$. Suppose that $f$ is bounded on some neighbourhood of $c$ show by example that if $\lim_{x\rightarrow c}g(x)$ ...
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3answers
181 views

A finite dimensional vector space that is not naturally isomorphic to its dual.

I need an example of a finite dimensional vector space $V$ that is not naturally isomorphic to $V^\ast$. I know that, at least in finite dimensional case, there is a one-to-one correspondence between ...
2
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0answers
97 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?
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34 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 ...
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1answer
56 views

A question on spaces with calibre-$\aleph_1$

Suppose that $X$ is the $T_1$ space with $k$-in-countable base and $\aleph_1$ is a caliber of $X$. Must $X$ be second countable? Thanks for any help. A topological space has calibre $\aleph_1$ if ...
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1answer
47 views

Counterexample to $\operatorname{Supp} \mathcal G \subset Z \hookrightarrow X$ but $\mathcal G\cong i_*i^{-1}\mathcal G$ when $Z$ is not closed

Let $\mathcal G$ be a sheaf on a topological space and $X$ (say a sheaf of sets) and suppose its support is contained in a subset $Z$ of $X$, i.e. $\operatorname{Supp} \mathcal G \subset Z ...
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1answer
65 views

Another question on second countable spaces

Let $X$ have countable chain condition and point countable base. Is $X$ second countable? I thing it don't need to be. However I have no examples at hand. Thanks for your help.
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69 views

Seeking counterexample or proof for equivalence of two separation properties

Two points $x,y$ of a topological space are said to be distinguishable if at least one has a neighborhood not containing the other, and separated if each has a neighborhood not containing the other. ...
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1answer
89 views

A counter example about fraction of 2 sets

In a software company there are 2 departments, called A and B. In department A, the fraction of senior programmers (of out the junior programmers in this department) finishing the jobs in time is ...
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1answer
321 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
4
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1answer
84 views

Examples of spaces in which the closure of any path connected set is path connected.

Are there (non-trivial) examples of topological spaces in which the closure of any path connected set is path connected? If so, are there any far reaching topological consequences of this property? ...
5
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1answer
2k views

Construct a monotone function which has countably many discontinuities

I read in a textbook, which had seemed to have other dubious errors, that one may construct a monotone function with discontinuities at every point in a countable set $C \subset [a,b]$ by enumerating ...
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1answer
212 views

Explicitly showing cokernel of exponential sequence is not a sheaf

In the classical example of short exact sequence of presheaves $$0\rightarrow \mathbb Z \rightarrow \mathcal O_X \rightarrow \mathcal F \rightarrow 1,$$ it is well known that $\mathcal F$ is not a ...
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1answer
83 views

Counterexample to in $\mathcal{Mod}_A$ colimits of filtered index categories are exact

This is not a true general fact for any abelian category, as Vakil points out in 1.6.12. He gives the following counterexample, which puzzled for it is in the category of abelian groups, and every ...
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1answer
176 views

The product of limit point compact Hausdorff spaces is not limit point compact

Let $X, Y$ be limit point compact Hausdorff spaces (to be clear, a space is said to be limit point compact if every infinite subset of it has a limit point). Is it true that $X \times Y$ is limit ...
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2answers
189 views

If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

Suppose that $R$ and $S$ are unital rings and that $S$ is a subring of $R$ in the weak sense where the multiplicative identities $1_R$ and $1_S$ are not assumed to be the same. In fact, assume $1_R ...
2
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1answer
46 views

AB does not imply KC

We say that a space $X$ is: 1)AB provided that $X$ is $T_1$ and for each pair $A,B$ of compact disjoint subsets of $X$ there is $U$ an open subset of $X$ such that either $A\subseteq U$ and $U\cap ...
0
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1answer
39 views

If $V$ is finite-dimensional and exist $\beta$ basis of $V$ such that $T(\beta)$ is a basis for $W$, then $T$ is a isomorphism?

Let $V$ and $W$ vector space over $F$ and $T : V \rightarrow W$ lineal. The statement is false, but I can't find a counterexample.
2
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1answer
180 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
3
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1answer
82 views

Can we find an example of non-mesuarable set which their outer measure could be computed?

We know there is non-measuarable set and we know every set has outer measure, so can anyone give me an example of a non-measuarable and there outer measure could be computed ?
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2answers
87 views

A non-UFD where we have different lengths of irreducible factorizations?

A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then ...
2
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1answer
84 views

Good concrete examples for understanding the different notions of monomorphism

In category theory, there's many variants on the notion of "monomorphism," such as: split monomorphism effective monomorphism regular monomorphism strong monomorphism extremal monomorphism What ...
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1answer
72 views

Examples of sets which measure cannot be obtained by discretisation

I started reading "An introduction to measure theory" by Terence Tao. On page 23 on a pdf reader (pg 7 in the actual document), we are asked to think of an example of a set $E\subset$ ...
5
votes
1answer
516 views

Tensor products over field do not commute with inverse limits?

In the question: Inverse limit of modules and tensor product, Matt E gives an example where inverse limits and tensor products do not commute over the base ring $\mathbb{Z}$. He then goes on to show ...
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1answer
158 views

Principal stable $SL(2)$-bundles on a genus $2$ compact Riemann surface.

Let $X$ be a compact Riemann surface with genus $2$. Can you give me examples of stable principal $SL(2)$-bundles on $X$?
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2answers
111 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 ...
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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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2answers
572 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 ...
3
votes
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 ...
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0answers
406 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
votes
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 $f(z)$ ...
1
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1answer
49 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
382 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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1answer
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 $X$. ...
0
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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 ...
5
votes
1answer
288 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
votes
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
69 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
420 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 ...
3
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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
108 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
votes
2answers
634 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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2answers
452 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 ...