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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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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101 views

Is there a well known nontrivial counterexample to this claim?

Suppose we have $A\subseteq\mathbb{N}$ with the property that if $B\subseteq\mathbb{N}$ and $B$ is finite, then $\exists a\in A\setminus\{ 1\}$, $\forall b\in B$, $\gcd(a,b)=1$. Are there any well ...
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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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39 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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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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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 $\lim f(x)/g(x)$ ...
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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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4answers
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Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality

Let $f: A\longrightarrow B$ be a function. 1)Prove that for any two sets, $C,D\subseteq A$ , we have $f(C) \setminus f(D)\subseteq f(C\setminus D)$. 2)Give an example of a function $f$, and sets ...
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Do we have always $f(A \cap B) = f(A) \cap f(B)$?

Suppose $A$ and $B$ are subsets of a topological space and $f$ is any function from $X$ to another topological space $Y$. Do we have always $f(A \cap B) = f(A) \cap f(B)$? Thanks in advance
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3answers
858 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?
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306 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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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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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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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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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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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 ...
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1answer
66 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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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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42 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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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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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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211 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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$z_0$ 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
628 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
60 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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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
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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
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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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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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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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107 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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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
157 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 ...
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1answer
60 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 ...
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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
31 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
86 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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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 ...
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1answer
183 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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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 ...
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2answers
284 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
342 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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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
89 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
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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 ...
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0answers
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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 ...
351
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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
139 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 $(0,1)$ ?