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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0answers
39 views

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 ...
2
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2answers
71 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 ...
1
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1answer
118 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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3answers
62 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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2answers
15 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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4answers
105 views

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 ...
2
votes
1answer
31 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 ...
1
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1answer
40 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), ...
4
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1answer
48 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 ...
0
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1answer
53 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
29 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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30answers
13k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
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3answers
55 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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2answers
64 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
28 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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1answer
19 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
28 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
33 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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1answer
3k views

Intersection of finite number of compact sets is compact?

Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true. I said that this is true because the intersection of ...
0
votes
1answer
64 views

Product of quotient maps and quotient space that is not Hausdorff

I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff.
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1answer
29 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, ...
0
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1answer
31 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, ...
2
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1answer
49 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 ...
3
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1answer
97 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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0answers
303 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
2
votes
1answer
41 views

Example of a ring $R$ such that $R\otimes_R R\not\simeq R$

I want to show the following statement: If a ring $R$ is commutative and $I,J\triangleleft R$, then $$ R/I\otimes_R R/J\simeq R/(I+J). $$ I can easily show this, assuming that $1\in R$. What can be a ...
0
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1answer
88 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 ...
3
votes
1answer
46 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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votes
2answers
44 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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1answer
101 views

Is every subgroup of a simple group is itself simple [closed]

Is the statement True of false? Every subgroup of a simple group is itself simple.
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1answer
300 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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6answers
4k views

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 ...
4
votes
1answer
77 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 $f$ ...
0
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1answer
55 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?
4
votes
1answer
62 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 ...
2
votes
2answers
42 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 ...
2
votes
1answer
97 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 ...
153
votes
31answers
9k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
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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 ...
1
vote
1answer
37 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 ...
0
votes
2answers
65 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 ...
8
votes
2answers
2k views

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
77 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, ...
2
votes
1answer
24 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 ...
1
vote
2answers
80 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.
2
votes
1answer
54 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 ...
1
vote
2answers
62 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 ...
0
votes
1answer
44 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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1answer
26 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 ...