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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Objects with a “Homogeneity Principle”

So I don't have to worry about formalities, in the following let $\mathscr{C}$ be a sufficiently nice category--at least nice enough so that the following definition makes sense. I believe concrete ...
3
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1answer
78 views

Does Self Duality imply Hamiltonicity?

I just looked at some self dual graph examples in the web and found that all of them are hamilton. Are there non-hamiltonian self dual graphs, or does self duality imply hamiltoncity? I have no ...
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1answer
551 views

Two variable limits via paths - are there pathalogical examples?

In the first year calculus course at my university, we do not introduce the $\varepsilon$-$\delta$ definition of a limit. When considering the limit of a function of two variables, we resort to paths. ...
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2answers
2k views

If $f$ is Lebesgue measurable on $[0,1]$ then there exists a Borel measurable function $g$ such that $f=g$ ae?

If $f:[0,1]\to\mathbb{R}$ is Lebesgue measurable then there exists a Borel measurable function $g:[0,1]\to\mathbb{R}$ such that $f=g$ a.e.?
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4answers
152 views

$\int f_k\to 0 $ but $f_k $ does not converge to $0 $ ae, where $ f_k $ is defined in $[0, 1] $

Give an exemple, in [0, 1], of a sequence of functions $ f_k $ such that $||f_k||_ 1=\int |f|_k \to 0 $ but $ f_k $ does not converge to $0 $ a.e.
5
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1answer
190 views

Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?

I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds: Example 9.3. Let $\Gamma$ be the graph of the function $f(x) = \sin(1/x)$ on the ...
2
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3answers
423 views

sequentially continuous on a non first-countable

Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this ...
10
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1answer
366 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
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4answers
2k views

Famous uses of the inclusion-exclusion principle?

The standard textbook example of using the inclusion-exclusion principle is for solving the problem of derangement counting; using inclusion-exclusion (and some basic analysis) it can be shown that ...
8
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2answers
254 views

Examples of closed subspaces of Baire spaces that fail to be Baire?

I am looking for some nice examples of Baire spaces containing closed subspaces that fail to be Baire. Clearly, $X$ should not satisfy either one of the standard hypotheses for the Baire category ...
2
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1answer
118 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
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1answer
117 views

A counter-example of the second isomorphism theorem for topological groups

Let $G$ be a topological group and $H$ and $N$ subgroups. Suppose $H$ is contained in the normalizer of $N$, then by using arguments of the second isomorphism theorem we can show that there is a ...
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1answer
184 views

Counterexample for finite dimensional weak convergence

Could you give an explicit construction of a sequence $\mathbb P_n$ of probability measures on $C[0,\infty]$ which converges in the sense of finite dimensional distributions BUT does not converge ...
8
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1answer
400 views

The perception of mathematics

In my work I wrote the following sentence. "...there is a negative perception of mathematics and mathematicians, both within and outside of academia." Err. Right. So, I believe that this is ...
6
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2answers
148 views

How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?

In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to ...
0
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0answers
55 views

Counter-example of an affirmation [duplicate]

Does anyone know any example that invalidates the following affirmation: If a morphism $f:A\to A$ induces the identity $\hat f:\operatorname{Spec} \left( A \right) \to \operatorname{Spec} \left( A ...
4
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1answer
180 views

Outer measure discontinuous from below

I was trying to find an example of an outer Measure which is not continuous from below. These are the definitions I use An outer measure on $X$ is a function $\mu^\ast: \mathcal{P}(X)\to ...
0
votes
1answer
351 views

Examples of homeomorphisms between the real numbers and the positive real numbers?

I'm interested in homeomorphisms between the real numbers, $\mathbb{R}$, and the positive real numbers, $(0,\infty)$--where both spaces have the topology induced by the metric $d(x,y)=|x-y|$. Here ...
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4answers
146 views

Infinity and structures

Do you know any case (example) where an "infinite" object with a structure (say, an infinite group) cannot be extended (in the sense of adding elements) in any way without it no longer having the ...
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2answers
130 views

Must certain rings be isomorphic to $\mathbb{Z}[\sqrt{a}]$ for some $a$

Consider the group $(\mathbb{Z}\times\mathbb{Z},+)$, where $(a,b)+(c,d)=(a+c,b+d)$. Let $\times$ be any binary operation on $\mathbb{Z}\times\mathbb{Z}$ such that ...
0
votes
2answers
362 views

locally compact Hausdorff space which is not second-countable

I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't ...
2
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4answers
181 views

Example of a module for non-mathematicians

I'm looking for a non-trivial1 example of a module that would be recognizable to a non-mathematician. I.e. I'm looking for examples of modules that one may come across in "the real world". The ...
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4answers
256 views

Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$

Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$? That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$?
0
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1answer
121 views

Lim sequence $\neq$ lim subsequence

Let $\{x_n\}$ be a sequence and $\{y_k\}$ be a subsequence of $\{x_n\}$. Can you show me an example of $\{x_n\}$ such that $\displaystyle \lim_{k \to +\infty} y_k= +\infty$, but $\displaystyle \lim_{n ...
5
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3answers
533 views

Does the Laplace transform biject?

Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.' Can someone provide a proof or counterexample ...
2
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2answers
90 views

Is the set of all distinct mathematical number types countable?

I was reading this article where the author explains that there are numbers outside the complex set and that you can arbitrarily generate new types using the same method as he described to generate ...
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votes
1answer
465 views

A simple example of Lindelöf space.

Somebody can to give me a simple example of Lindelöf space? Note. Lindelöf space is a topological space in which every open cover has a countable subcover.
3
votes
1answer
840 views

Regular space which is not Hausdorff

I know that normality in the absence of $T_{1}$ does not imply regularity (Sierpinski space being a counterexample as it is vacuously normal but not regular). I have the feeling that similarly ...
0
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5answers
90 views

Operator with symmetric but without associative?

Addition and multiplication in math, all is symmetric,associative。 But i have no idea about operators with symmetric but without associative. please help me listing any 2-arity operators?
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3answers
126 views

Ring and Subring with different Identities [duplicate]

Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know ...
2
votes
1answer
122 views

Where is the fallacy in this coupling argument of two Bernoulli variables?

With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function: $$\begin{align*} f(\mathcal{J}) = ...
7
votes
3answers
571 views

Banach Algebra counterexample

Can someone give me an exemple of a Banach Algebra $\mathbb{A}$, for which there is no isometric representation in a Hilbert Space ? (with proof or references to proof) Thank you very much :)
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2answers
1k views

Is there a function with infinite integral on every interval?

Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
3
votes
2answers
173 views

Sequence convergence and parentheses insertion

find an example for a series $a_{n}$ that satisfies the following: $a_{n}\xrightarrow[n\to\infty]{}0$ ${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ does not converges There is a way to insert ...
4
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3answers
2k views
3
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2answers
140 views

Constructing a separable space that is not hereditarily separable.

The construction that I had in mind was: Let $X$ be an uncountable space. We'll assume it has just one countable dense subset $E$. Let points $p_{1},p_{2}\in X\setminus E$ such that every open set ...
1
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1answer
68 views

Is every perfectly normal space submetrizable?

In the previous question, Some counterexamples are given which shows shat not every perfectly normal space is paracompact. Thanks Henno. I have another question. It may be difficult: Is every ...
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1answer
153 views

Is a perfectly normal space always a paracompact space?

Is a perfectly normal space always a paracompact space? A topological space $X$ is called a perfectly normal space if $X$ is a normal space and every closed subset of $X$ is a $G_\delta$-set. ...
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votes
2answers
241 views

Surjective and continuous map between Hausdorff spaces

Can we say that a surjective and continuous map $p:X\to Y$ is a quotient map iff both $X$ and $Y$ are Hausdorff? If not, could you give me an example? I am terrible at finding counterexamples.
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1answer
54 views

Example ideal of $\mathfrak{sl}(2,\mathbb{C})$

I need an example about ideal from lie algebra $\mathfrak{sl}(2,\mathbb{C})$ except trivial ideal and $\mathfrak{sl}(2,\mathbb{C})$ itself, can someone help me? I try to make ideal except trivial ...
3
votes
1answer
150 views

On groups whose center has odd order

Let $G$ be a finite group such that $Z(G)$ is of odd order and $Inn(G)$ is of even order. Then prove $G\simeq Z(G)\times N$, such that $N$ is a subgroup of $G$ where $N\simeq Inn(G)$. Thank you!
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2answers
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Real world well formulated examples of non linear optimization problems

I'm trying to find around the web some real world examples of non linear optimization problems. I currently need examples of: Non restringed optimziation ( $\max$/$\min$ $f(x)$ for ...
6
votes
2answers
168 views

Infinitely many zeros of a nonconstant continuous function?

Let $f:[0,1]\to\mathbb{R}$ be a nonconstant continuous function. Is $S=\{x: f(x)=0\}$ finite? I have thought of a function with countably many $0$'s like lots of triangular bumps at each point ...
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1answer
185 views

An example of Kirchhoff equation

I'm studying about a simply kind of Kirchhoff equation in one-dimensional that means $$\begin{cases} ...
8
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2answers
786 views

Continuity of the inverse $f^{-1}$ at $f(x)$ when $f$ is bijective and continuous at $x$.

Prove or disprove: Let $f:\mathbb{R}\to\mathbb{R}$ be bijective and $f$ is continuous at $x$. Then $f^{-1}$ is continuous at $f(x)$. Any hints are welcome. If this is false, I would like to have ...
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2answers
83 views

prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ when $n$ is odd

let $n$ be odd integer , prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ it's an example which the text proves ! but i can't understand any thing from the argument ! but i tried to ...
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1answer
168 views

Category-theoretic cross product and set-theoretic cross product

I recently proved as an exercise the associativity of cross product as defined in category theory. But in set theory, cross product is not associative. It seems intuitive to me that cross should be ...
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3answers
135 views

Is this kind of space metrizable?

It has a nice result from Tkachuk V V. Spaces that are projective with respect to classes of mappings[J]. Trans. Moscow Math. Soc, 1988, 50: 139-156. If the closure of every discrete subset of a ...
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1answer
113 views

A question on metrizable space

This exercise is from "General Topology" by Engelking: Give an example of a metrizable space which cannot be embedded in a locally compact metrizable space. I don't how to start. The ...
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1answer
68 views

Understanding an example of M. L. Wage, W. G. Fleissner, and G. M. Reed

In this paper of M. L. Wage, W. G. Fleissner, and G. M. Reed, the authors claimed that having a zeroset diagonal does not guarantee submetrizable by showing Example 2. However, the example is very ...