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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Examples of magmas with all their elements idempotents

A magma is supposed to be closed under a binary operation. Are there examples of magmas with all their elements idempotents under the operation of the magma?
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Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Recently, someone stated that every short exact sequence (of, say, modules) of the form $$0 → M → M \oplus N → N → 0$$ splits. I think this is false in general because the arrow $M → M \oplus N$ might ...
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Examples of applications of mononotone and pseudomonotone operators

Hi I am aware that the following question is quite broad, but I would appreciate any feedback even if it is in the form a reference. I am interested in some standard examples in engineering (or any ...
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1answer
35 views

Does convolution of two functions in $H^s(\mathbb{R})$ belong to $H^{2s}(\mathbb{R})$?

Let $f$, $g$ be two density functions and assume that $f,g\in H^s(\mathbb{R})$, $s>\frac{1}{2}$, where $${H^s}(\mathbb{R}) = \left\{ {u:\int_{ - \infty }^\infty {{{\left| {\hat u\left( t \right)} ...
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83 views

Does $(X)(Y)=(XY)$ for $X,Y\subseteq R$?

Let $R$ be a commutative ring. Denote by $X\ast Y=\{xy\mid x\in X,y\in Y\}$ the complex product of subsets. I want to show that given subsets $X,Y\subseteq R$ the following ideals are equal: ...
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72 views

Is this series $\sum_{n \geq 2}\sqrt{a_n}\frac{n^{a_n}-1}{\ln n}$ always divergent?

Let $\displaystyle \sum a_n$ be a series with positive terms which is a convergent series and suppose that we don't have $\displaystyle a_n \ln n \rightarrow 0$. Is the following series always ...
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What is an example such that $f(x)\neq \sum_{m=0}^{k-1} \frac{f^{(m)}(\alpha)}{m!} (x-\alpha)^m + \frac{f^{(k)}(\psi)}{k!} (x-\alpha)^k$?

Let $f:[a,b]\rightarrow \mathbb{C}$ be a $C^{k-1}$ and assume $f^{(k-1)}$ is differentiable on $(a,b)$. If the range of $f$ is real, then the usual taylor's theorem holds, but I'm not sure whether ...
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1answer
32 views

Topological Spaces: Pre-Uniform Structures

Disclaimer This thread is meant to record. See: Answer own Question Reference It is a follow-up to: Uniform Spaces: Neighborhood System It has relevance to: TVS: Uniform Structure Problem Given ...
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Two disjoint connected and bounded open sets in the plane that shares the same boundary

In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary. My attempt: Since both my sets need to be bounded, my ...
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Topological Spaces Involving Connectedness, Compactness, and Hausdorfness

I made a table about topological spaces with or without connected, compact, and Hausdorff properties. However, I cannot find the example for the following cases: Compact, but neither Hausdorff nor ...
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An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
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1answer
35 views

Is every open set in a base space evenly covered?

Let $C$ be a covering space of $B$. Then, does every open set in $B$ evenly covered by a covering map? This must be false but I cannot find a counterexample.. Please help
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1answer
33 views

commuting subsets in a group

For a countable infinite discrete group $G$, consider the following three properties. (P1) $G$ is abelian. (P2) For any finite subset $K$ of $G$, there exists an element $s\in G$, such that the two ...
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0answers
64 views

Counterexample to an implication

Denote $\bar{A}$ a complement of $A$ in a set $\Omega$ and $A \Delta B = A/B \cup B/A$ the symmetric difference of $A, B$. It is claimed that for a map $\phi := \Omega \rightarrow \lbrace 0, 1 ...
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Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
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Find two functions $f$ and $g$ such that they are both discontinuous at $c$, however, $f+g$ and $f\cdot g$ are both continuous at $c$

Could someone please explain to me how to approach these kinds of question and also what is the answer to the following question? Give an example of a function $f$ and $g$ such that they are both ...
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Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
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2answers
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The product of two nonnegative, improperly integrable functions is also improperly integrable.

True or False: The product of two nonnegative, improperly integrable functions is also improperly integrable. I was given both the problem and the proof that may or may not be true. I think the ...
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2answers
41 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
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1answer
24 views

Show that the space of superharmonic functions is not a linear space

Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $P=(p_{i,j})_{i,j\in E}$. A real valued function $h$ on $E$ is called superharmonic if $h(x)\geq Ph(x)$ ...
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1answer
44 views

Open (but not closed) subgroups of $GL_n$

The book I am currently reading states: "...as we will see later, non-closed subgroups [of $GL_n(\mathbb K)$] are not necessarily manifolds." Prompted me to think about open subgroups of $GL_n$: ...
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1answer
22 views

Lebesgue integrable discontinuity points

If a function is Lebesgue integrable, is it possible that it has as set of discontinuity points measure bigger than zero?
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1answer
40 views

Arrange 10 points on five lines where each line(intersecting) has exactly 4 points

One possible case is that forming a star and then arranging 10 points on its vertices. Is there any other possible case for this arrangement? If not then how can we prove it mathematically? ...
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1answer
49 views

What is a counterexample for this one?

Let $x$ be an irrational number. Let $\{a_0\}$ be the sequence of positive integers except for $a_0$ such that $x=a_0+K(1/a_n)$. Let $a,b$ be integers such that $b>0$ and $gcd(a,b)=1$ and ...
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3answers
100 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
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1answer
47 views

An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$

I am thinking about this problem: Let $f\in L^1 [0,1]$ to be a nonnegative function satisfied: $$\int_{E} f dm\leq \sqrt{m(E)}$$ for every measurable set $E\subset [0,1]$, Prove that $f\in ...
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0answers
19 views

Why is it so difficult to obtain the spectral properties related to infinite matrices, especially when they are not symmetric?

I believe most of the spectral theory is revolving around the bounded self-adjoint linear operators being analogous to real symmetric infinite matrices. Whereas, there are cases when the matrices are ...
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1answer
42 views

Compactness of second-countability of $\omega$X$\omega_1$

Please discuss the following properties of the product space consisting of $\omega$X$\omega_1$: Is it compact? Is it 2nd countable? $\omega$ is the first infinite ordinal and $\omega_1$ is the ...
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3answers
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Complex Mean Value Theorem: Counterexamples [closed]

This thread is just to collect some examples... Given an open domain $\Omega\subseteq\mathbb{C}$. Consider a holomorphic function $f:\Omega\to\mathbb{C}$. What would be a counterexample to: ...
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1answer
90 views

Are the following topological spaces locally compact?

I am trying to determine whether the following spaces are locally compact: a) the slotted plane b) the radial plane For part a) I am almost certain that it is not compact, but not sure how to go ...
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1answer
52 views

Counterexample to conditional probability with dependent events

Let $X1,X2,X3$ be i.i.d. taking values in a finite set, and not constant. Is it necessarily true that $P(X3=X2|X2≠X1)≤P(X3=X2)$? Give a proof or a counterexample. Since the two events $A=\{X3=X2\}$ ...
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1answer
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$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Let $F$ be a field and $K$ be an extension field of $F$. Proof\Counterexample: $K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$. I haven't ...
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Unconventional (but instructive) proofs of basic theorems of calculus

Inspired by this questions asked on MathOverflow, I would like to ask if you know some "sophisticated" proofs of the basic theorems in a calculus course (that is, the ones that you can find, for ...
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2answers
30 views

Non-unital commutative semigroups $S$ such that for all $x \in S$, the function $S \rightarrow S$ given by $y \mapsto x+y$ is a bijection?

Does there exist a commutative semigroup $S$ with the following (additively denoted) properties? For all $x \in S$, the function $S \rightarrow S$ given by $y \mapsto x+y$ is a bijection. $S$ has no ...
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0answers
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Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
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1answer
20 views

Example of an increasing non-nonnegative sequence violating conclusion of monotone convergence theorem in space of finite measure

With Lebesgue measure in $\mathbb{R}$, $f_n(x) \equiv -\frac{1}{n}$ is a good example which doesn't coincide with MCT. However, I couldn't find another example when the measure is finite. Could ...
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1answer
82 views

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this ...
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0answers
29 views

Necessary and sufficient condition

could someone help me with 'necessary and sufficient' conditions problem with full proofs or counterexamples. thank you. It's either a 'necessary', 'sufficient', 'necessary and sufficient' or ...
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1answer
67 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
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1answer
43 views

Cantor Intersection Theorem Without Closedness, counterexample

The Cantor Intersection Theorem is that Let $\{S_1,S_2,S_3,...\}$ be a countable collection of nonempty sets in $\mathbb R$ such that: $S_{k+1} \subset S_k$ for $k=1,2,3...$ Each $S_k$ ...
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Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$?

How to determine the existence of the function $f: \mathbb{R} \rightarrow \mathbb{R}$, which is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$? It's more than enough to give an ...
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1answer
35 views

Metrizable compact spaces and Hausdorff spaces with a countable network

I have two questions related to metrizable spaces and countable network ; Can we find a continuous mapping from a separable metric space onto a non metrizable compact Hausdorff space. If a Hausdorff ...
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2answers
56 views

A category with arbitrary products, but not all limits, or finite limits not commuting with filtered colimits?

I'm interested in finding an example of a locally small category $\mathcal{C}$ having small filtered colimits and arbitrary small products but lacking, either all small limits, or either the ...
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2answers
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Why does differentiability implies continuity, but continuity does not implies differentiability?

Why does differentiability implies continuity, but continuity does not implies differentiability? I am more interested in the part about a continuous function not being differentiable. Well, all ...
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0answers
10 views

Iterated limits difficult example

Is there a fucntion $f:[-1,1]\subseteq \mathbb R^2 \to \mathbb R$ so that $$lim_{x\to 0}f(x,y)$$ exist $\forall y_0\in [-1,1]$ fixed and and $$lim_{y\to 0}f(x,y)$$ exist $\forall x_0\in [-1,1]$ ...
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2answers
24 views

Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $lim_{x\to x_0}f(x,y)$ does not exist

Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $$lim_{y\to y_0}f(x,y)$$ and $$lim_{x\to x_0}(lim_{y\rightarrow y_0}f(x,y))$$ exists but $lim_{x\to x_0}f(x,y)$ does not ...
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Flow-Invariance w.r.t. $A$ doesn't invariance w.r.t. $\exp (tA)$

This answered question of mine explains that by saying consider $$\dot x =Ax,$$ then the stable and unstable subspaces are invariant with respect $A$ and therefore also with respect to $\exp (tA)$ one ...
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3answers
88 views

Need example for a topological space that isn't connected, but is compact

We know the topological space $(R,T1)$ is a connected space but it is not compact, (R,T+) (which generated by [a,b]) is not connected space and it is not compact space, and $(R,Tcf)$ is connected and ...
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1answer
55 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...