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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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
24 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 ...
3
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2answers
55 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 ...
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3answers
62 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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10 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 ...
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26 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
26 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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3answers
724 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 ...
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1answer
17 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 ...
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1answer
34 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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2answers
62 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
56 views

Double sequence, two sequences converge, but to different limits?

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, ...
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0answers
55 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
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1answer
49 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 ...
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2answers
65 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
19 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
23 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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1answer
25 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
41 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
45 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
21 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 ...
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3answers
174 views

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 ...
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1answer
28 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 ...
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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 ...
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1answer
25 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
42 views

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 ...
0
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0answers
29 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
72 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 $\lim f(x)/g(x)$ ...
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1answer
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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2answers
33 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?
4
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1answer
55 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
119 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 ...
3
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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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2answers
29 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 $i_0,i_1 \in \Bbb Z$ ...
0
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1answer
26 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→+∞ ...
3
votes
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 ...
0
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1answer
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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votes
3answers
95 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
36 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
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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0answers
19 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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1answer
57 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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0answers
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: ...
4
votes
1answer
50 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 ...
5
votes
1answer
46 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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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 ...
12
votes
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 ...
4
votes
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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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
votes
0answers
33 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 ...