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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An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open.

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
5
votes
2answers
58 views

Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
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0answers
22 views

Example of Non-Measurable Sets in Product Space

If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\...
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1answer
15 views

Simple example of a function which is in $W^{1,p}(\Omega)$ but not in $L^{\infty}(\Omega)$?

I am looking for a simple (intuitive) example of a function $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is an open set and, obviously, $p \leq N$. Sobolev embedding theorem asserts ...
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1answer
54 views

Regular topological spaces need not to be normal

I was looking for a counterexample for the following statement: "A regular topological space need not to be normal." I don't understand how to use the lemma to prove Theorem 7: http://fac.hsu.edu/...
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4answers
45 views

Find a counter example

The interior of the union is the union of the interiors. $\text{int}\left(A\cup B\right) = \text{int}(A) \cup \text{int}(B)$ I'm not too sure about to get started with this one. Any hints so as to ...
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0answers
28 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
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3answers
84 views

Are there any examples of Graphs in nature? [closed]

When it comes to fractals, there are several examples we can point to and say 'this is a fractal', such as snowflakes, ferns, trees and coastlines. Are there any equally clear examples of graph and ...
8
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1answer
63 views

Is there a measure space $(X,\mathcal M, m)$ such that $\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$?

I have in mind the following question: Is there a measure space $(X,\mathcal M, m)$ such that the range of $m$ satisfies $S:=\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$? (...
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1answer
43 views

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$? Well, in the answer is no. it is written that $e^{x+y}$ for every $(x,y)$ has ...
3
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1answer
52 views

Is there a searchable database of mathematical objects that you can search by property?

For example, I could search for functions that are continuous, but that don't have differentiability, and come up with a continuous non-differentiable function. Or a smooth but non-analytical function....
4
votes
2answers
91 views

Is boundedness required in equivalence between $\frac1n\sum_{k=1}^na_k\to0$ and $\frac1n\sum_{k=1}^na_k^2\to0$?

Suppose $a_n$ is a sequence of non-negative real numbers. If $a_n$ are un-bounded, then I want to know if $\dfrac{1}{n}\sum_{k=1}^na_k\to0$ as $n\to\infty$ is equivalent to $\dfrac{1}{n}\sum_{k=1}^...
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0answers
12 views

Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ : $$ dist\big(\gamma(s), \gamma(t)\...
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0answers
35 views

Are there groups $G$ isomorphic to $\mathrm{Aut}\left( \mathrm{Aut} (G)\right)$, such that $G\not\cong \mathrm{Aut}(G)$?

Trivial examples for the first condition are easy to find: $G={1},C_2$. Are there any (finite\non-finite) groups that satisfy the conditions in the title?
2
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1answer
53 views

Define any non-commutative operation for the group $\left({\mathbb{R}, \circ}\right)$ [closed]

Let $\mathbb{R}$ denote the set of real numbers. Given that $\left({\mathbb{R}, \circ}\right)$ is a group, provide any definition for $\circ$, so that $\circ$ is not commutative.
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3answers
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A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
-1
votes
4answers
70 views

Disprove the statement: If $g\circ f=I_X$then $f\circ g=I_Y$. [closed]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: Let $f : X \rightarrow Y$ and $g : Y \rightarrow X$ be functions. If $g\circ f=I_X$...
4
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1answer
102 views

are continuous functions that map measure zero sets to measure zero sets absolutely continuous?

Let $I$ be a closed interval and $f:I\rightarrow \mathbb{R}$ be a continuous function which maps measure zero sets to measure zero sets. If $f$ is monotonically increasing, $f$ must be absolutely ...
2
votes
1answer
28 views

If a sequence converges in measure, are convergent subsequences of it all converge to the same limit?

Let $f_n:X\rightarrow \mathbb{C}$ be a sequence of measurable functions such that $f_n\rightarrow f$ in measure. Let $f_{n_k}$ be a subsequence of $f_n$ such that $f_{n_k}\rightarrow g$ pointwise a.e....
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4answers
1k views

Non-abelian group with infinitely many abelian subgroups

I'm looking for a non-abelian group which has infinitely many abelian subgroups. Do you know any examples of such groups?
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4answers
112 views

Disprove the statement $f(A \cap B) = f(A) \cap f(B)$ [duplicate]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: If $f : X \rightarrow Y$ is a function and $A$, $B$ are subsets of $X$ then $f(A \cap B) = f(A)...
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2answers
31 views

Why can projection function on $X \times S$ be regarded as a local homeomorphism?

I am studying some properties of local homeomorphism I am in particular trying to find a local homeomorphism that is not a homeomorphism and the projection function seems to be the perfect candidate ...
2
votes
1answer
76 views

Domain Monotonicity - Neumann eigenvalue problem (Edit)

Related to the question : http://mathoverflow.net/questions/242136/why-m-1-subset-m-2-not-rightarrow-n-m-1-lambda-leq-n-m-2-lambda The Neumann eigenvalues of the rectangle with sides $a$ and $b$ are $...
1
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1answer
35 views

Does there exists two way embedding between two non-homeomorphic spaces?

I am searching for a two way embedding between two non-homeomorphic spaces. In other words, I want two non-homeomorphic spaces such that $X$ is embedded in $Y$ and $Y$ is embedded in $X$. Recall ...
1
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1answer
28 views

Reference request: product Borel $\sigma$-algebra of non-separable metric spaces

The following is a proposition in Folland's Real Analysis about product sigma algebra: Here $\mathcal{B}_X$ denotes the Borel $\sigma$-algebra on $X$. Could anyone come up with an example that ...
1
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2answers
55 views

Can spaces where all singletons are closed and all singletons are open be homeomorphic?

Suppose $(X, \mathfrak{T})$ is a space where all singletons are closed, and $(Y, \mathfrak{J})$ is a space where all singletons are open. Can these two spaces be homeomorphic? My thought is that ...
0
votes
1answer
54 views

If $\limsup(na_n) = 1$, then $\sum\limits_{n=1}^{\infty} a_n$ diverges

Let $a_n$ be a sequence of positive numbers. Suppose $\limsup(na_n) = 1$. Does this mean $\sum a_n$ diverges? I have only concluded this if the limit superior is in fact the limit of the sequence. ...
3
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2answers
83 views

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is not continuous?

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is not continuous?
1
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1answer
41 views

Example highlighting the difference between finer and strictly finer?

For instance, what does it mean to say that the lower limit topology on $\mathbb{R}$ is strictly finer than the usual topology on $\mathbb{R}$? I understand why lower limit topology is finer. Take ...
3
votes
1answer
43 views

Monoids in which $aN = Na$ and $ab \in N \leftrightarrow ba \in N$ aren't equivalent.

Proposition. Let $G$ denote a group, and $N$ denote a subset of $G$. Then the following conditions are equivalent: $aNa^{-1} = N$ $aN = Na$ $ab \in N \leftrightarrow ba \in N$ Proof. ...
2
votes
0answers
40 views

Implications of disproving the Goldbach's Conjecture

What would be the most important implications of finding an even number that cannot be expressed as the sum of two primes? Would the existence on one such number in anyway predict the likeliness of ...
1
vote
1answer
31 views

Example of a strictly convex function unbounded in $\mathbb{R}$

Is there some strictly convex function defined in $\mathbb{R}$ to be unbounded(above and lower)? For example, $f:(\infty,0]\to \mathbb{R},$ $f(x)= -x^2$ is a strictly convex function. However, this ...
0
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3answers
61 views

If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$

I wish to prove whether this is true or false. If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$. I'm not even sure if $N$ being normal ...
4
votes
2answers
92 views

Element of a ring without unity which divides every other element

Question. Is there an example of a ring $R$ (commutative or not) without unity and an element $x \in R$ such that for every $y \in R$ there exists a $z \in R$ such that $y = x z$? In other words, is ...
1
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0answers
22 views

Example of a semimartingale with special properties

I have to find an example of a semimartingale X such that $\lim_{t \rightarrow \infty} X_t$ exists a.s. and $X$ is not a semimartingale up to infinity. I think it could be a deterministic function ...
0
votes
1answer
62 views

About commutative ring with identity

Let $R$ be an infinite commutative ring. Which of following options is false? Center of $M_2(R×R)$ is nontrivial. $ M_2(R×R) \cong M_2(R)×M_2(R)$ The number of units in $M_2(R ×R)$ is infinite. The ...
0
votes
2answers
28 views

Prove or disprove : limit point compact hausdorff space imply compact space?

Prove or disprove : limit point compact hausdorff space imply compact space?. I think that it is not true. Because we know that the every product of compact space is compact and we know that product ...
3
votes
2answers
118 views

A discontinuous function with smooth sections

I am searching for $f : U\rightarrow \mathbb R $ defined in an open square $U$ in $\mathbb R^2$ so that $(0,0) \in U$, $f$ is not continuous at $(0,0)$, for each $x$ the function $y\mapsto f(x,y)$ is ...
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0answers
14 views

Specific examples of Side Information?

I'm starting to apply information theory to gambling. There is something called Side information (see details in [1]), which I understand is additional information about the outs of the game. It could ...
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votes
1answer
78 views

Is it possible to have a homomorphism from an infinite group to a finite group? [closed]

Why my book say that "it is possible to have a homomorphism from an infinite group to a finite group"? How can I prove it?
3
votes
2answers
106 views

If $g$ is uniformly continuous and $f(x)$ is close to $g(x)$ (for large enough $x$), is then $f$ uniformly continuous?

Suppose $f$ is continuous on $(0,\infty)$ and $f$ is simmilar $g$ for all $x>M$ ($M>0$). (i.e. for any $\epsilon >0$, there is $M>0$ such that if $x>M$, then $|f-g|< \epsilon$) is ...
0
votes
0answers
39 views

Normal extension of a field

Let $F$ be an extension of $K$ (they are both fields). I know that if $F$ has finite degree over $K$, then the following things are equivalent: 1) $F$ is such that every irreducible polynomial in $K[...
3
votes
1answer
33 views

What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
0
votes
1answer
129 views

Prove or disprove $2abc(a+b+c)\ge 3(a^2b^2c^2+1)$

Let $a,b,c>0,ab+bc+ca=3$, prove or disprove $$2abc(a+b+c)\ge 3(a^2b^2c^2+1)$$ Now I can't find any counterexample
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1answer
75 views

Continuous or Differentiable but Nowhere Lipschitz Continuous Function

What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval? What is a real valued function that is differentiable on a close interval but not ...
1
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1answer
35 views

A graph with list chromatic number $4$ and chromatic number $3$

What is an example of a graph with chromatic number $\chi(G)=3$ and list-chromatic number $\chi_\ell(G)=4$? My first thought was to consider complete tripartite graphs since these will have chromatic ...
0
votes
1answer
24 views

Torsion elements and subgroups of nonabelian group

I am currently studying torsion groups and I am playing around with defintions to get used to them. An element $g \in G$ is a torsion element, if there exists $n \in \mathbb{N}$ so that $g^n = e$, ...
0
votes
1answer
18 views

Example of an uncountable metric space where every point is isolated

I was trying to come up with an example of an uncountable metric space all of whose points are isolated. I've had difficulty thinking of one, has anyone got any nice examples? Just in case: ...
1
vote
1answer
68 views

Examples of the use of $(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$ in “real” mathematics?

Here, a proof is requested for the following tautology: $$(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$$ Its pretty easy to prove; nonetheless, I don't find the formula at all intuitive, ...
12
votes
2answers
420 views

Does an abelian subgroup inject into the abelianisation of the whole group? [closed]

If $H <G $ are groups and H is abelian, do we get an injection from H into $G/[G,G] $?