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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3
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1answer
56 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....
0
votes
1answer
57 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/...
3
votes
1answer
1k views

$[0,1)\times[0,1)$ (lower limit topology) is a regular, but not a normal topological space

Let $X=[0,1)\times[0,1)$, $\tau$ its topology with base $$\beta = \{ [a,b)\times[c,d): 0 \leq a < b \leq 1, 0 \leq c < d \leq 1 \}\;.$$ Please help me prove, that it is regular, but not a ...
42
votes
10answers
13k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
1
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3answers
90 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 ...
0
votes
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 ...
1
vote
0answers
33 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 ...
8
votes
1answer
80 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 $S:=\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$? (...
1
vote
1answer
46 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 ...
2
votes
1answer
54 views

Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
4
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2answers
93 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}^...
4
votes
2answers
481 views

Example of a topological vector space such that $E = M \oplus N$ algebraically, but not topologically

I have the following question: Give an example of a topological vector space $E$ with subspace $M$ and $N$, such that $E = M \oplus N$ algebraically, but not topologically (so $E \ncong M \sqcup N$...
0
votes
1answer
136 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
16
votes
1answer
402 views

Find $f(x,y )$ such that $f_{x},f_{y},f_{yx}$ are continuous,but $f_{xy}$ is not

Let $f$ be a function of two variables,let$(a,b)$ be a point and let $D$ be an open disk with center $(a,b)$. Assume that $f$ is $\mathcal C^{1}$ on $D$, and $f_{yx}$ exist on $D$. Further, the ...
0
votes
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?
3
votes
1answer
64 views

core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
6
votes
3answers
110 views

Examples of non-abstract rings?

In group theory, it helped me a lot to use symmetry groups of geometrical objects like a triangle to understand the more abstract concepts. Are there rings with a similar low level of abstractness? I ...
6
votes
1answer
132 views

Prove or disprove that ${F_{n}^2} + 43$ is always a composite

This is a kind of follow-up to another question, but in order not to burden that question and its answers with new comments, I decided to create this separate question. Also, it looks this problem is ...
2
votes
1answer
54 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.
10
votes
3answers
1k views

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 ...
4
votes
1answer
109 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 ...
-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 $g\circ f=I_X$...
11
votes
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?
2
votes
1answer
29 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....
33
votes
5answers
2k views

Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
3
votes
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)...
-1
votes
1answer
84 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?
1
vote
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
77 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 $...
3
votes
2answers
107 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 ...
1
vote
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
vote
1answer
29 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
vote
2answers
57 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 ...
-1
votes
1answer
59 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. ...
1
vote
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
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?
3
votes
1answer
44 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
42 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 ...
1
vote
0answers
72 views

Counterexample for “Filtration of stopping time equals filtration generated by stopped process”

I am working in a discrete setting. Consider any stochastic process $(X_n)_{n\in\mathbb N}$ with its natural filtration $(\mathcal F_n)_{n\in\mathbb N}$ and a stopping time $\tau$. We know that $$\...
3
votes
1answer
46 views

$f_{\scriptscriptstyle{\vert H}}=g_{\scriptscriptstyle{\vert H}}$ implies $f=g$ for groups

Is it possible to find a group $G_0$ and a proper subgroup $H$ such that for all morphism $f,g$ from $G_0$ to $G_1$ such that $f_{\scriptscriptstyle{\vert H}}=g_{\scriptscriptstyle{\vert H}}$ implies $...
4
votes
2answers
96 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 ...
0
votes
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 ...
1
vote
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 ...
32
votes
3answers
1k views

If every continuous $f:X\to X$ has $\text{Fix}(f)\subseteq X$ closed, must $X$ be Hausdorff?

Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$. In a recent comment, I wondered whether $X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is closed for every continuous $f:X\...
4
votes
4answers
176 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
2
votes
4answers
177 views

Assume that the function $f(x)$ is continuous and $\lim_{n\to\infty}f_n(x)=f(x)$. Does this imply that $f_n(x)$ is uniform convergent?

I wonder if this is true: Let $(f_n)$ be a sequence of real-valued functions defined on a set $S\subset\mathbb{R}$. Assume that the function $f(x)$ is continuous and $\lim_{n\to\infty}f_n(x)=f(x)$. ...
2
votes
2answers
2k views

Example of a general random variable with finite mean but infinite variance

Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, ...
3
votes
2answers
121 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 ...
0
votes
1answer
63 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 ...