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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Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
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2answers
81 views

Example of $\pi$-metrizable space

A tychonoff space $X$ is $\pi$-metrizable if and only if it has a $\sigma$-locally finite $\pi$-base. Please help me to find some example of $\pi$-metrizable space. Is it true that every ...
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602 views

Total Variation Measure: Definition?

The total variation according to Rudin is defined as: $$|\mu|(E):=\sup_{\bigcup_{k\in\mathbb{N}}E_k=E}\sum_k|\mu(E_k)|$$ where the supremum is taken over all countable partitions. Now I'm reading in ...
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76 views

Borel Sets which are not intervals

I am looking for an element of the Borel-sigma-algebra which is not an (open, closed, half-open,...) interval. Can someone provide any example or an algorithm to construct them?
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27 views

What would be an example that rank of a subgroup of a free group is greater than the rank of free group?

Let $G$ be a free group and $H$ be a subgroup of $G$. Then, $H$ is also a free group. Let $R_G,R_H$ be ranks of $G,H$ respectively. what would be an example that $R_H>R_G$?
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What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
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1answer
39 views

Constructing noncommutative nilpotent rings of given index

When I read about algebra I often see a certain disregard for examples or perhaps a disregard for a reader whose knowledge of examples is limited. When I'm interested in a property $p$ of an algebraic ...
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1answer
119 views

Atlases on the topological manifold $\mathbb R$

I have been trying to produce an example of two incompatible atlases on $\mathbb R$. But no success. Could someone help me please? All my example seem compatible. For example, $A_1 = \{((-\infty,1), ...
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60 views

One-sided and two-sided cancellability

Is there a semigroup $S$ with right- and left-cancellable elements but no elements cancellable from both sides? $s\in S$ is left-cancellable if for any $a,b\in S$ we have $$sa=sb\implies a=b$$ and ...
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1answer
63 views

Seeking a possible counterexample in probability.

I am trying to find a counterexample or prove the following: $\dfrac{Var\left(X_{n}\right)}{\left[EX_{n}\right]^{2}}\rightarrow0 , then \dfrac{X_{n}}{EX_{n}}\rightarrow1$ in probability. Assuming ...
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1answer
65 views

Product Hilbert Spaces that require completion?

My lecturer said that to make the Hilbert space $(H\oplus H',⟨\cdot,\cdot⟩ )$ we need to (1) make the cartesian product $H \oplus H' = H\times H'$, (2) give it an inner product, and - the confusing ...
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79 views

Does a symmetric matrix $A^2$ imply a symmetric $A$?

Does a symmetric matrix $A^2$ imply a symmetric $A$? Any help would be much appreciated.
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153 views

Suppose f : A ---> B and g : B ---> A are functions for which g o f = 1A…

If I were to suppose that $f : A \to B$ and $g : B \to A$ are functions for which $g \circ f = 1_A$, is $f$ always surjective and is $g$ always injective? How would I either prove this or counter it? ...
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1answer
31 views

Continuity and Subspace Topology

I think the first one is false. If we let $(-1/2, 1/2) \subset \Bbb R$ and $(0,1/4) \subset \Bbb R$, then for $f(x) = x$ defined on $[0,1) \subset M = \Bbb R$, we have $f^{-1}(-1/2, 1/2) = ...
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65 views

A net in a product space and its cluster point

Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces. And let $Z=X\times Y$ be the product space equipped with the natural product topology $\mathcal{T}_Z$ on $Z$. Then, let ...
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1answer
44 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
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1answer
44 views

Ring subset which absorbs but is not an additive subgroup

Are there any undegrad-level examples of ring subsets which possess absorbtion property (as in ideal definition) but are not ideals (i.e. are not additive subgroups)?
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56 views

Example of ring with two maximal ideals such that the char of the quotients is $0$, respectively $p$.

I am looking either for an example of a commutative ring with identity and two maximal ideals, such that the characteristic of one of the quotient rings is finite and the other characteristic is zero, ...
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1answer
33 views

Stuck trying to find unbounded $s_n$ with $\frac{1}{n}\sum_{k=1}^n s_k\rightarrow L$

I proved that if a sequence $(s_n)$ converges to a limit $s$ then so does its "average sequence," $(\sigma_n)$ with $\sigma_n=\frac{1}{n}\sum_{k=1}^n s_k$. I found a counterexample for the converse, ...
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1answer
138 views

Proving the derived set $E'$ is closed.

I was reading the proof in Rudin, but it uses the metric. Is this not true if $X$ is a general topological space and $E' \subset X$ (especially if it is not Hausdorff $T_1$)? I can't come up with a ...
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1answer
107 views

intersection of locally compact Hausdorff topologies.

Are there locally compact Hausdorff topologies $\mathcal T, \mathcal S$ on a set $X$, such that $\mathcal T\cap \mathcal S$ is a Hausdorff but not locally compact topology on $X$?
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1answer
116 views

How to construct a smooth curve whose range is dense in $\mathbb R^2$?

How to construct a smooth curve $f: \mathbb R \to \mathbb R^2$ whose range is dense in $\mathbb R^2$? Space-filling curves are well-known, but they cannot be smooth. The image of a smooth ...
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1answer
78 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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2answers
59 views

Example of differentiable function with $f'(s_{n})=0$ but $f'(0)>0$

Ex: Give an example of a differential function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $0$ is a limit (accumulation) point of a sequence of critic points ($f'(x)=0$) and however $f'(0)>0$ ...
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585 views

Examples of Baire class 2 functions

Do you know of examples of Baire class 2 functions which are not Baire class 1 functions, besides the the indicator function of the rationals and the indicator function of the Cantor set?
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How are eigenvectors/eigenvalues and differential equations connected?

In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...
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1answer
183 views

A derivative which is not Lebesgue integrable on any interval?

If $f=x^2\sin(x^{-2})$, then $f'$ exists everywhere (including $x=0$) but $f'$ is not Lebesgue integrable on $[0,1]$ (precisely because of the singularity at $x=0).$ I'm trying to find a function $f$ ...
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1answer
104 views

Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
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1answer
429 views

Are primitive row stochastic matrices diagonalizable?

Let $A$ be an $n \times n$ matrix with real, non-negative entries. Assume $A$ is primitive, meaning there exists an integer $k$ such that $A^k>0$ (here the inequality means all entries in $A$ are ...
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2answers
66 views

Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$

Let $f$ be a mapping. Prove that the following three statements are equivalent. $f$ is continuous; $\forall A \subseteq X: f(\overline{A}) \subset \overline{f(A)}$; $\forall B \subseteq ...
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1answer
112 views

Splitting of Nonmeasurable Sets

Being curious I'm wondering: Let $V$ be a Vitali set defined as usually as a choice of $v\in[r]$ with $0\leq v\leq 1$ for every $[r]\in\mathbb{R}/\mathbb{Q}$. Since the countable disjoint union of ...
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0answers
26 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
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1answer
55 views

A Real valued function which is discontinuous **only** on a given specific set.

Let $\mathbb{L}=\{x_n \ |\ n=1,2,3 \dots\}$ be a countable subset of $\mathbb{R}$. My aim is to construct a real valued function $f$ on $\mathbb{R}$ such that $f$ is discontinuous at every point ...
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2answers
104 views

Vector space without a scalar product

In linear algebra the terms vector space and scalar product always (at least for me) appear together. Can you give me an example of a vector space without a scalar product? Does the senescence Let V ...
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Is there an example of a sigma algebra that is not a topology?

Is there an example of a sigma algebra that is not a topology? If this is not the case, is it possible to prove that all sigma algebras are topologies?
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2answers
133 views

Can the derivative of an absolutely continuous real function have a simple discontinuity?

If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous). More specifically, ...
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1answer
54 views

Finding a $C^1$ function $F(x,y)$ that is not of class $C^2$

I am trying to find a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ of class $C^1$, but not of class $C^2$. Meaning that $\frac{\partial^2 F}{\partial x \partial y}$ won't be equal to ...
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2answers
115 views

A perfect Hausdorff space that is not metrizable.

Can anyone provide an example of a well-known topological space that has the following three properties: (1) It is perfect (contains no isolated points), (2) T2, and (3) not metrizable.
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1answer
95 views

Sets and Logic .. Disproving with counter example

Can anyone please give me an idea to disprove the following with a counterexample: $A , B , C$ are sets. If $A \times C = B \times C$ , then $A = B$. (Here $\times$ is a Cartesian product.) I tried ...
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2answers
103 views

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges?

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges? $x_n$ is said to be bounded if and only if it is bounded both above and below. I believe this to be false. My ...
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1answer
125 views

Sobolev embedding counterexample

I trying to find a counterexample to show that $$W^{1,p}(\mathbb{R ^n}) \nsubseteq C^{0,\alpha}(\mathbb{R^n}) $$ for $p>n$ and $\alpha > 1 -\frac{n}{p}$. No clue yet, thanks for your help.
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38 views

A function with only a partial derivative not Hölder-continuous

I'm looking for a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$ 1. $x\mapsto u(t,x)$ is $C^{2,\alpha}$; 2. $t\mapsto u(t,x)$ is $C^{1,\alpha}$; 3. $t\mapsto ...
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Examples of “finite character”

The property of being a linear ordering has finite character, i.e. a relation linearly orders a set if it linearly orders all of its finite subsets. This is a trivial corollary of the definition. ...
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48 views

Can analytic function be represented as a taylor series?

Let $D$ be open in $\mathbb{K}$. Let $f:D\rightarrow \mathbb{K}$ be an analytic function. Then, $f$ is infinitely differentiable and $\forall x_0\in D$, there exists a neighborhood $N$ of $x_0$ such ...
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1answer
110 views

Question on Taylor's theorem

Taylor's Theorem Let $\{c_n\}$ be a sequence of complex numbers. Let $R$ be the radius of convergence of $\sum c_n z^n$. Let $|b|<R$ and $f(x)=\sum c_n z^n$ on the disk $B(0,R)$. ...
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584 views

Does there exist a non-abelian group with roots?

I just learned the definition of a division group: An abelian group $G$ is called divisible if for every $x\in G$ and every $k\in\mathbb{Z}^+$, there exists $y\in G$ such that $y^k=x$ (or $ky=x$, ...
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2answers
41 views

A Tychonov space non-homogeneous wrt neiborhood bases cardinality

Is there a Tychonov space $(X,\mathcal T)$ with $a,b\in X$ such that $a$ has a countable neighborhood basis while $b$ does not have any countable neighborhood bases?
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103 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
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1answer
108 views

Examples of reduced associative algebras

An associative $K$-algebra A is called reduced if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. ...