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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1answer
39 views

Can we construct a continuous but nowhere differentiable surface?

In my analysis course we learned about the Weierstrass function which is continuous but nowhere differentiable, is it possible to make a surface which is continuous and nowhere differentiable?
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26 views

An example of a prime quotien ideal where his corresponding ideal is not prime

If $R$ is a comutative ring with identity ring and $K$ is an ideal from it, let $R'=R/K$ and $I$ an ideal of $R$ satisfy $K\subseteq I$ and $I'$ is the coresponding ideal of $R'$ (we knew that ...
4
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3answers
68 views

Trivial power of line bundle

I'm trying to understand the following: Thinking of a line bundle as a bunch of locally generating sections together with transition functions (which in this case are just multiplication by local ...
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3answers
75 views

Is there a countably compact sequential non-$T_2$ space that is not sequentially compact?

Let $X$ be a topological space. Definitions: $X$ is countably compact if every countable open cover of $X$ has a finite subcover or equivalently, every sequence in $X$ has a cluster point. $X$ is ...
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1answer
61 views

Example of graph with specific $\chi (G)$, $\omega (G)$, $\beta (G)$

Find an original example of a graph whose chromatic number does not equal its clique number, yet whose clique partition number equals its independence number. Chromatic number: $\chi(G)$ is the ...
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545 views

Example of a ring with an infinite inclusion chain of ideals [closed]

I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)
3
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1answer
122 views

Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either: Proofs i.e. pointwise equicontinuity is uniform ...
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0answers
50 views

Greedy algorithm fails to give chromatic number

Produce a graph and degree sequence for which the greedy algorithm fails to give the chromatic number. My first example is below- The first labeling uses 2 colors which is the chromatic number and ...
0
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1answer
49 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
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1answer
36 views

An example of smooth compactly supported function with non-vanishing Fourier transform

I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the ...
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1answer
40 views

Example of maximum modulus principle

As it's known , an holomorphic($\neq constant$) function $f:G\subseteq\mathbb{C}\rightarrow \mathbb{C}$ has maximum modulus on $\partial G$ . I wuold an example of a function holomorphic on a disk ...
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133 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
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5answers
544 views

A Compact Hausdorff Space with no Manifold Structure? [closed]

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
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0answers
30 views

What are types of coalgebras that are more naturally described by cooperads?

Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of endomorphisms and ...
3
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3answers
104 views

What exactly is the maximal solution of an ODE and why do we care?

I am reading these notes on the definition of a maximal solution of an ODE i.e. http://www.math.lmu.de/~philip/publications/lectureNotes/ODE.pdf But the definition is so abstract and no example is ...
4
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0answers
62 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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1answer
39 views

An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and $\text{diam}(...
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87 views

Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it? Though there are ...
3
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3answers
82 views

Example of Topological Group

I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology. ...
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0answers
15 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
0
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0answers
28 views

Prove or Disprove If $T : R^n \rightarrow\ell_2$ is linear, then $T(A)$ is totally bounded in $\ell_2$ when $A ⊆ R^n$ is bounded

Prove or Disprove, If $ T: \mathbb R^n \rightarrow \ell_2 $ is linear, then It preserves total boundedness $ T(A) $ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded. I think ...
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1answer
36 views

problems to understand a special definition of “free graded commutative algebra” from lecture

I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., ...
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797 views

Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
0
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1answer
32 views

Why is semi-locally simply connected defined this way?

I would like to know why we define a space $X$ to be semi-locally simply connected if $\forall p\in X \exists U\ni p: i(\pi_1(U))=0\subset \pi_1(X)$ (SLSC), where $i$ is induced by $U\...
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1answer
45 views

Example of a sequence with at least 3 limit points [closed]

What is an example of a sequence that has at least 3 limit points?
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71 views

Example of a ring for which $rs \neq 0$ but $sr = 0$. [duplicate]

I am looking for an example of an associative noncommutative ring $R$ with the following property: for $r,s \in R$, $$ rs \neq 0, \text{ but } sr = 0. $$ Moreover, do rings for which this cannot ...
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2answers
73 views

Double sequence, two sequences converge, but to different limits? [duplicate]

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, n})...
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0answers
74 views

Counterexamples in integral calculus: do functions like these exist?

Could you give me examples of functions of the following kinds? A function which is Riemann-integrable AND has an antiderivative, but is not continuous A function which is Riemann-integrable, and ...
0
votes
1answer
59 views

Pseudocompact space having countable $\pi$-base but not metrizable

Can we find a normal Hausdorff space which is a countably compact locally connected space without isolated points and has a countable $\pi$-base but not metrizable? A collection $\mathcal{B}$ of ...
2
votes
5answers
114 views

If $\{x_n\}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?

Since the definition of Cauchy sequence is: Understanding the definition of Cauchy sequence, I noticed we need an absolute value for $a_m-a_n$ in the definition so the statement would be false. But I ...
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2answers
70 views

If $A\dot{-} B$ is countable and $B \dot{-} C$ is countable then $A\dot{-} C$ is countable? [closed]

Prove that: If $A\dot{-} B$ is countable and $B\dot{-} C$ is countable then $A \dot{-} C$ is countable? If not give a counter-argument
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1answer
40 views

Does anybody know an example for a matrix with nullspace property for nonnegative signals?

A vector is $k$-sparse, if there are at most $k$ non-zero entries. In compressed sensing an arbitrary matrix $A\in\mathbb{R}^{m\times n}$ (with usually $m<n$) is said to have the null space ...
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0answers
28 views

False Counterexample for “for all sets A, B, and C, A ∩ (B - C) = (A ∩ B) - (A ∩ C)”

I've put together a proof on this, (which I would appreciate being verified), but I also want to know what a false counterexample might be for this? I'm new to discrete mathematics, and I'm honestly ...
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3answers
98 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
0
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1answer
56 views

Limit point Compactness does not imply compactness counter-example

I think that I understand why compactness implies limit point compactness: Suppose $A \subseteq X$ has no limit points. Then $A^{\prime} \subseteq A$. Thus, $A$ is closed. Then for all $a \in A$, ...
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1answer
47 views

What are some interesting cases where the two obvious definitions of “discrete object” diverge?

The nLab page defines "discrete object" as follows: Definition. [nLab] Let $\mathbf{C}$ denote a concrete category whose forgetful functor $U$ has a left adjoint $F$. Call the counit of this ...
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1answer
22 views

Equality of two multilinear forms

Take two multilinear forms $f,g$ defined on the same set $E$ such that $\forall x\in E,f(x,x,\dots,x)=g(x,x,\dots,x)$. Does that imply that the two functions are necessarily equal ? I can't seem to ...
5
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counterexample to a “theorem” on continuity of largest deltas for continuous functions $f:[a,b]\to\mathbb{R}$

"Theorem 12" in these notes states the following (verbatim): Let $f:[a,b]\to\mathbb{R}$ be continuous and let $\epsilon>0$. For $x\in[a,b]$, let $$\Delta(x)=\sup\left\{\delta\,\,|\,\,\text{...
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1answer
43 views

Questions on symmetric difference of events

From a comment on my math overflow question: No, $P(A\bigtriangleup B)=0$ means $A$ and $B$ are essentially the same except in situations that almost surely do not happen. $P(A)=P(B)$ says much ...
3
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1answer
45 views

$f_{\scriptscriptstyle{\vert H}}=g_{\scriptscriptstyle{\vert H}}$ implies $f=g$ for groups

Is it possible to find a group $G_0$ and a proper subgroup $H$ such that for all morphism $f,g$ from $G_0$ to $G_1$ such that $f_{\scriptscriptstyle{\vert H}}=g_{\scriptscriptstyle{\vert H}}$ implies $...
29
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10answers
2k views

Non-associative commutative binary operation [duplicate]

Is there an example of a non-associative, commutative binary operation? What about a non-associative, commutative binary operation with identity and inverses? The only example of a non-associative ...
0
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1answer
30 views

Can't the first (/second) transinfinite ordinal replace the first (/second) uncountable ordinal in several counterexamples?

An example of a sequentially compact but not compact space is $\omega_1$. Indeed, any sequence of countable ordinals either has infinite elements below some countable ordinal, or has an ascending ...
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2answers
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strict local extremum of $f'$ that is neither saddle nor inflection value of $f$

Is there a function $f$ with the following properties: $x_0$ is a strict local extremum of $f'$. $(x_0,f(x_0))$ is neither a saddle point of $f$ (i.e. a point with $f'(x_0) =0$ which is not local ...
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0answers
32 views

What's the biggest number used as a counterexemple? [duplicate]

I'm looking for exemples of big numbers that are counterexemple of some interesting conjecture. Do you know conjectures that seemed to be true until a million (or many more) numbers where checked?
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1answer
97 views

counterexample relating to l'Hopital's rule

Suppose there are two funtions $f(x),g(x)$ such that (as $x \to a$) we have $f(x) \to +\infty$, $g(x) \to +\infty$, and $f'(x)/g'(x) = g(x)/f(x)$. Then by l'Hopital's rule, if $\lim f(x)/g(x)$ exists,...
0
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1answer
51 views

Connectedness and path connectedness of a union

Exercise: Let $A = \{(x, \sin (1/x)): 0<x\le 1\}$ and $B = \{(x,y)\in\mathbb R_{\le 0}\times\mathbb R | 0.5\le |y|\}$ be sets and $X = A\cup B$ the union. Show that $X$ is connected and path ...
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2answers
41 views

Open and connected set in metric space [duplicate]

In a normed space, we know that if a set is open and connected, it is path connected. Is it true for general metric space or general topological space?
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1answer
64 views

Is every Hausdorff homogeneous space also regular?

Every Hausdorff topological group is regular (completely regular, in fact). Is this true if I replace topological group with homogeneous space? This is not obvious to me because there are Hausdorff ...
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2answers
156 views

Sequence in $l^p$ but not $l^q$ for all $q<p$

I need to find a sequence for real $p>1$ so it is in $l^p$ but not in any of the space $l^q$ with $1 \leq q <p$. I tried the sequence $(1/n)^{1/q}$ which is in $l^p\setminus l^q$. However, this ...
1
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3answers
924 views

Function with infinitely many right inverses?

I was thinking about a function with infinitely many right inverses but I could not come up with anything. Does there exist a function with infinitely many right inverses?