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
310 views

A strongly pseudomonotone map that is not strongly monotone

Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that $F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that $$ \left<F(y)-F(x), ...
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2answers
145 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 ...
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2answers
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Example of Converge in measure, but not converge point-wise a.e.?

Can anyone give an exam of Converge in measure, but not converge point-wise a.e.? And also for the converse part, professor asks us to prove "pointwise a.e. implies converge in measure", but think ...
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2answers
33 views

Suppose $d$ is a common divisor of two numbers $a_0$ and $a_1$, can $d$ be written as a linear combination of $a_0$,$a_1$ with integer coefficients?

Suppose $d$ is a common divisor of two numbers $a_0$ and $a_1$, can $d$ be written as a linear combination of $a_0$,$a_1$ with integer coefficients? i.e. there exists two integers $i_0,i_1 \in \Bbb Z$ ...
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1answer
28 views

Construct an example where x(t, x_0) is bounded but limt→+∞ x(t, x_0) does not exist.

Suppose we are given an IVP $x = f(x), x(0) = x_0 $, x ∈ R^n for which we know that the unique solution x(t, x_0) exists globally in time. Construct an example where x(t, x_0) is bounded but limt→+∞ ...
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2answers
512 views

How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?

Are the floor functions of $0.999\cdots$ and 1 equal? It is true that $0.999\cdots=1$ but how does one justifies the integer part of $0.999\cdots$ being 1 , where it is not, or alternatively without ...
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7answers
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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 ...
3
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1answer
69 views

Does there exist a Hausdorff group which is not locally compact?

A topological space is countably compact if every countable open cover has a finite subcover. A topological space $X$ is locally compact if any point has a neighbourhood which is compact. A ...
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1answer
112 views

Is a path-connected bijection $f\colon \Bbb{R}^n \to \Bbb{R}^n$ continuous?

While thinking about this question I was asking myself if a path-connected bijection $f\colon \Bbb{R}^n \to \Bbb{R}^n$ has to be continuous for $n>1$? If we drop the requirement that $f$ is ...
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1answer
53 views

Examples and counter-examples for rings

Here's what I am trying to do: Listing mnemonics used: $D_n(\Bbb R)$ to denote diagonal $n \times n$ matrices with real coefficients. I.D. - Integral domain, E.D. Euclidean Domain Much of the ...
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3answers
103 views

If G is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in R. is it true in general metric space?

If $G$ is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in $\mathbb R$. is it true in general metric space? I know as $G$ is open and singleton set ...
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1answer
40 views

Example of two analytic functions that differ at countably infinity many point

$\displaystyle f_1(x) = \frac{x^n-1}{x-1}$ and $f_2(x) = x^{n-1} + \cdots + 1$ have the same values except at $x=1$ (where $f_1$ fails to be analytic ). Is there an example of two analytic function ...
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3answers
247 views

Example of an uncountable dense set with measure zero

As stated in the title, I am trying to find an example of an uncountable dense subset of $[0,1]$ that has measure zero. My intuition is that such a subset cannot exist, but I do not have a proof of ...
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0answers
21 views

$z_0$ is an essential singularity of $f$-> is $z_0$ is an essential singularity of $\frac{1}{f}$?

Let $f:B_{\epsilon}(z_0)\setminus \{z_0\}\to \mathbb{C}$ holomorphic, without zero points and $z_0$ is an essential singularity of $f$. Question: Does $\frac{1}{f}:B_{\epsilon}(z_0)\setminus ...
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2answers
682 views

The range of the derivative of a differentiable function

I read somewhere that, given a function $f$ differentiable on $[a,b]$, the range of $f'$ can be (1) a closed interval or (2) an open interval or (3) a half-open interval or (4) an unbounded interval ...
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1answer
67 views

Spans of subsets and union of two sets

Is this true or not? How would I prove or disprove this? If the set of vectors $\{a_1 \dots a_n\}$ spans a subset $S$ and the set of vectors $\{b_1 \dots b_n\}$ spans a subset $T$, then ...
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1answer
77 views

Example of a non-abelain finite group $G$ with $G/N$ abelian and infinite group $G$ with $G/N$ finite

Have not been able to think of a examples with the following properties: Example of a non-abelian finite group $G$ with property that $G/N$ is abelian for every non-trivial normal subgroup $N$ of ...
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0answers
47 views

Filters and their refinements vs nets and their subnets [duplicate]

True or false? a) Let $(x_{\alpha})_{\alpha\in A}$ a net over a space $X$ and $(x_{h(\beta)})_{\beta\in B}$ a subnet, where $h:B\to A$ is monotone and final. Let $\mathcal{F_1}$ be the filter ...
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1answer
21 views

Determining if a set is measurable by upper and lower sets

I have the following question regarding Lebesgue measure: If $A,B$ are measurable sets and I have $m(A\setminus E)=0$ and $m(E\setminus B)=0$, is it enough to determine that $E$ is measurable? We do ...
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1answer
83 views

Example of a subnet that have no subsequence.

I have an elementary question on nets because I'm not familiar with this concept. Here are two basic facts: Every subsequence of a sequence is a subnet; Not every subnet of a sequence is a ...
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1answer
170 views

Give an example of a program in Simply Typed Lambda that produces Bottom.

I'm not sure how bottom applies to simply typed lambda calculus. not A is a common abbreviation for A -> ⊥ But I see no way to construct a function of that signature within the theory. Edit: A more ...
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0answers
111 views

I feel like this cannot be proven. Am I setting up the contrapositive correctly?

The question ask: Use proof by contrapositive to show that if a positive integer is the product of  two distinct primes, then its square root is irrational. So I have not(q) -> not(p) as follows: ...
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1answer
47 views

Recreational conjecture on factoring groups

Consider the following: For a group $G$ with identity $e$, define $s: G \to \mathbb{N} \cup \{ \infty \}$ by $s(g) = \min \{ k \in \mathbb{N} : g^{k} = e \}$, where $ \min \emptyset = \infty$. ...
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1answer
174 views

How to show “If A and B connected, is $A\cup B$ connected”?

If A and B connected, is $A\cup B$ connected? or give a counterexample. I'd say no because when we take $A=[1,2]$, $B=[3,4]$, these closed intervals are connected. But when we take $U=]\frac ...
2
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1answer
63 views

Examples where $R/I\cong R$? [duplicate]

I had to prove on a test that if $R$ is a PID then every surjective endomorphism of $R$ is an injection. To do this, I supposed there was a surjective endomorphism $\varphi:R\to R$. Then ...
4
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2answers
143 views

Prove or Disprove there is a sequence $f_n$ of continuous function on [0,1] such that for each x $ \in [0,1] $, $f_n(x)$ converges to $f(x)$

Prove or disprove: If $f$ is non-decreasing real valued function on $[0,1]$ then there is a sequence $f_n$ of continuous function on $[0,1]$ such that for each x $ \in [0,1] $, we have $f_n(x)$ ...
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2answers
37 views

Measure on a countable set

Is there a decent characterization of measure on an infinite countable set? At page 7 of "Introduction to Measure Theory and Integration" (Ambrosio, Da Prato, Mennucci), example 1.10 I found that ...
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1answer
103 views

Example of a Borel measure, which is not Borel-regular

I have asked a question to find four types of outer measures here, and I could find three of the four examples. We call an outer measure $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ Borel, if ...
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3answers
340 views

Function $f$ which isn't smooth but $f^3$ is smooth

In Pugh's Real Mathematical Analysis there is an exercise, marked with three stars (which denotes that the author doesn't know the answer), whether there exist a nonsmooth function $f : \mathbb{R} \to ...
12
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1answer
187 views

If $R[x]$ and $R[[x]]$ are isomorphic, then are they isomorphic to $R$ as well? [duplicate]

There are examples of commutative rings $R \neq 0$ such that $R[x]$ is isomorphic to $R[[x]]$ (see this question; an example would be $R=S[x_1, x_2, \ldots][[y_1, y_2, \ldots]]$, with $S \neq 0$ any ...
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3answers
937 views

What is the largest set for which its set of self bijections is countable?

I recently came across a problem which required some knowledge about the self bijections of $\mathbb{N}$, and after looking up how to construct some different bijections I came across the result that ...
3
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2answers
302 views

Is it true that every normal countable topological space is metrizable?

I've been reading about and working on various proofs about metrizabililty. I'm having trouble answering the following question: Is it true that every normal countable topological space is metrizable? ...
3
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1answer
368 views

Why is a monoid with right identity and left inverse not necessarily a group? [duplicate]

This problem is from Herstein's 'Topics in Algebra'. I've thought about it a bit but haven't come up with much. Let $G$ be a non-empty set with an associative product which also satisfies: $\exists ...
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3answers
2k views

Is a semigroup $G$ with left identity and right inverses a group?

Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group? Now, If both the identity and the ...
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1answer
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Any concrete example of ''right identity and left inverse do not imply a group''? [duplicate]

In the abstract algebra class, we have proved the fact that right identity and right inverse imply a group, while right identity and left inverse do not. My question: Are there any good examples of ...
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3answers
97 views

Finding an example of a set $G$ which is not a group

Suppose $G$ is a set and $\cdot$ is a binary operation on $G$ such that there exists an $e\in G$ such that $a\cdot e=a$ for a in $G$ and given $a\in G$, there is a $y(a)\in G$ such that $y(a)\cdot ...
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1answer
19 views

Metric space with two similar points which are not in the same orbit.

Is there an example of a metric space $X$ with two points $p$ and $q$ so that for every $r>0$ the ball with radius $r$ and center $p$ is isometric to the ball with radius $r$ and center $q$ and yet ...
3
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0answers
34 views

What could be examples at calculus or introductory analysis level for the idea contained in the statement by David Hilbert?

I read the following quote in the book "As opposed to abstraction the art of doing mathematics consists in finding special cases which contain all the germs of generality. --David Hilbert", however ...
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33answers
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Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
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1answer
144 views

A map from $(0,1)$ to $(0,1)$ such that the image of every open interval in $(0,1)$ is $(0,1)$

Can we have a map from $(0,1)$ to $(0,1)$ such that the image of every open interval in $(0,1)$ is all of $(0,1)$ ?
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1answer
15 views

Valid method to obtain a basis of a topological subspace?

Let $(X,\tau)$ be a topological space and $Y \subset X.$ We know that if $\mathcal{B}$ is a basis for $\tau$ and $\tau_{\small{Y}}$ is the subspace topology on $Y$, then we can obtain a basis for ...
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6answers
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Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
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1answer
75 views

Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...
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1answer
40 views

Definite integrals that are hard using the FTC but doable from first principles

When I was introduced to integration, I was briefly exposed to the definition of a Riemann sum, and as an illustration we worked out a couple of definite integrals directly from this definition: $$ ...
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1answer
34 views

Looking for example $\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f \,d\lambda$ where $\mu(A) < \infty$

I'm looking for a summable non-negative function $f: \Bbb{R} \to [0,\infty)$ and a measurable set $A$ with finite measure such that $$\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f ...
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1answer
120 views

Counterexample of polynomials in infinite dimensional Banach spaces

I'm trying to prove exercise I.3.B in Mujica's "Complex analysis in Banach spaces". DEFINITIONS: A map $P$ is an m-homogeneous polynomial from $E$ to $F$ if there is a m-linear map $A$ from ...
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1answer
30 views

Show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}})$ is in Sobolev space $W^{1,p}(B_1(0))$

As part of my seminar this semester, I need to show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}}) \in W^{1,p}(B_1(0))$. I have shown that $f$ is indeed in $L^p$, but could use some help proving ...
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1answer
51 views

Product of RREF versus RREF of product

Let $A$ and $B$ be two matrices of arbitrary shape where the number of columns of $A$ is the same as the number of rows of $B$. Is it always true that $$\textbf{rref}[A]\cdot \textbf{rref}[B] = ...
9
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1answer
102 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R} ^2$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
3
votes
2answers
588 views

Example for a sequence of continuous functions which converge to a continuous function pointwise but not uniformly on a compact set.

My actual aim is to verify that each of the conditions in Dini's theorem is essential or not.Theorem says that A sequence {$f_{n}$} of continuous functions defined on a compact set $E$ converges to ...