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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1answer
20 views

Both atoms and co-atoms in a lattice

Is there an example of a lattice (or just a poset) for which both atoms and co-atoms are useful?
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0answers
59 views

Maximum measurable collection and Caratheodory extension

Remember Caratheodory extension theorem: we get an outer measure by extending a premeasure over an algebra. We also have Caratheodory theorem, which give us a measure by restricting that outer measure ...
7
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1answer
1k 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 ...
3
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1answer
100 views

Give an example of a program in Simply Typed Lambda that produces Bottom.

I'm not sure how bottom applies to simply typed lambda calculus. not A is a common abbreviation for A -> ⊥ But I see no way to construct a function of that signature within the theory. Edit: A more ...
16
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2answers
650 views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
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1answer
50 views

Counter example for absolutely continuous measure

I need a example for the following statement: "Given a pair of finite measures $(\mu,\nu)$ on a given measurable space $(\Omega, \mathbb{A})$ is said to have property $P$ if for every $\epsilon >0$ ...
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1answer
43 views

To construct a $C^{\infty}$ function such that…

I'm trying to construct a $C^{\infty}$ function $f:\mathbb R \to \mathbb R$ such that $f=1$ on an interval $[a,b]$ and $f=0$ outside of some open interval $(a-\varepsilon,b+\varepsilon)$, for ...
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1answer
13 views

A criterion for complete lattice.

Is there an infinite partially ordered set $(X,\le)$, in which for each $A\subseteq X$, either $\inf A$ or $\sup A$ exists but for some $A\subseteq X$ either $\inf A$ or $\sup A$ does not exist.
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2answers
174 views

Examples of nowhere continuous functions

Do you know examples of nowhere continuous functions, besides the Dirichlet function and its modifications?
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1answer
697 views

Counterexample Math Books

I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in ...
3
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1answer
79 views

Does $\phi\circ g$ convex imply $\phi\circ f\circ g$ convex?

Assume that $\phi$ is an increasing function and $g$ is a decreasing function with $\phi\circ g$ convex. If $f$ is an increasing convex function does this imply $\phi\circ f\circ g$ is a convex ...
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0answers
46 views

Restricted continuity implies continuity

When teaching calculus, we instruct students to calculate multivariate limits using the following theorem: If $\gamma$ is a smooth curve with $\gamma(0) = a \in \newcommand{\R}{\mathbb{R}}\R^n$ ...
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1answer
75 views

Uniquely complemented lattice that is non-modular

I'm looking for an explicit example of a uniquely complemented lattice that is non-modular, since neither of the two non-modular lattices described here at wikipedia have this property. Thanks.
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2answers
72 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
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2answers
42 views

Significance of low order terms in base expansion of integer square root

My head is turning into a uniform gel of random thoughts! I cannot see a proof or find a counterexample to the following: Conjecture: Let integer $x$ be expressed as $a_3 \, b^3 + a_2 \, b^2 + a_1 \, ...
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1answer
158 views

A counter example of best approximation

Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm ...
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2answers
188 views

Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent

$p>1$ is a integer, Show a convergent series $\sum\limits_{n=1}^\infty a_n$, $a_n\in\Bbb R$, such that the series $$\sum_{n=1}^\infty a_n^p$$ is divergent p.s. If $p>1$ is not an integer ...
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2answers
151 views

A non-exponentially bounded analytic function?

A function $f:\mathbb R\to\mathbb R$ is said to be exponentially bounded if there is an $n$ such that for sufficiently large $x\in\mathbb R$, $\exp(\exp(\cdots \exp(x)))>f(x)$ (where the $\exp$ is ...
2
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1answer
49 views

Example of a function between boolean lattices that preserves $(\top,\bot,\wedge,\vee)$ but not complements.

Its easy to find boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ such that $f$ preserves both top and bottom elements, as well as binary meets, but not complements. ...
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0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
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4answers
90 views

Looking for a counter example - Normal subgroups and quotient group

I need to find a counter example for the following: Let $G$ be a group, and let $A,B\triangleleft G$ be two normal subgroups of $G$. if $G/A\cong B$ then $G/B\cong A$. Here are my thoughts so far: ...
5
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1answer
47 views

$L^1(\mathbb{R}^n)$ functions not in $\mathcal{H}^1(\mathbb{R}^n)$

I am wondering how to imagine the Hardy space on $\mathcal{H}^1(\mathbb{R}^n)$ and in particular what sort of functions are in $L^1(\mathbb{R}^n)\backslash\mathcal{H}^1(\mathbb{R}^n)$. Furthermore, is ...
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2answers
46 views

Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
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4answers
711 views

Example of a finite non-commutative ring without a unity

Give an example of a finite, non-commutative ring, which does not have a unity. I can't think of any thing which fits this question. I was thinking $M_2(\mathbb{R})$ but it has the identity. Any ...
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10answers
4k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
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1answer
38 views

An isometrie $\varphi: S_1\to S_2$ which cannot be extended into distance-preserving map

I'm searching for an example isometrie $\varphi: S_1\to S_2$($S_i$ are regular surfaces) which cannot be extended into distance-preserving maps $F: \Bbb R^3 \to \Bbb R^3$. A reference or hint will ...
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3answers
53 views

Non-trivial order on a set with infinitely many minimals/maximals

I am looking for an example of each: 1) A non-trivial order on a set A such that there are infinitely many minimal elements and 2) A non-trivial order on a set A such that there are infinitely many ...
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1answer
46 views

The Tangent Disc Topology is developable

A well-known example of a Moore space is the Tangent Disc Topology. I want to show that the Tangent Disc Topology is a developable space, i.e. it has a development. But I could not find the proof of ...
6
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3answers
92 views

Structures with addition, multiplication and exponentiation.

The set $\mathbb{N}$ can be viewed as a mathematical structure with operations off addition, multiplication and exponentiation. Observe that: It forms an Abelian monoid under both addition and ...
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0answers
178 views

An open problem on general topology

There is an open problem in this paper: J. van Mill, V.V. Tkachuk, R.G. Wilson, "Classes defined by stars and neighbourhood assignments", Topology and its Applications, Vol. 154, Issue 10, 2007, pp. ...
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1answer
52 views

Does there exist a suitable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n^m) = mf(n)$? I think no.

Any ideas how to prove that no injection $f : \mathbb{N} \rightarrow \mathbb{N}$ whose image is closed under multiplication by elements of $\mathbb{N}$ satisfies the following identity? $$f(n^m) = ...
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2answers
115 views

Do silly-rings exist?

A ring can be defined as a near-ring satisfying two-sided distributivity, whose underlying additive group is Abelian. Negating this second stipulation, we obtain the following definition. A silly-ring ...
2
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1answer
34 views

Examples of rational equivalence.

I have to do a lesson (1 h.) about a very basic introduction to intersection theory. In order to do this I'd like to find a way in order to examplain the concept of rational equivalence giving a lot ...
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4answers
3k views

Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?

I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of ...
3
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0answers
77 views

Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
40
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12answers
2k views

Examples of results failing in higher dimensions

A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging. One of the examples of rigor-lacking approach are proofs done via graphs or pictures ...
3
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1answer
38 views

Are there interesting examples of medial non-commutative semigroups?

There exist semigroups $S$ (written additively) such that $S$ is medial, meaning $(a+b)+(a'+b') = (a+a')+(b+b')$. $S$ is not commutative. Example. The left (and right) zero semigroups are all ...
2
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4answers
332 views

Examples of comma categories

There is a basic construction in category theory which I've only just recently become acquainted with, that is the comma category. It seems to be a quite basic construction for which, however, I've ...
3
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4answers
132 views

Number of elements in a group and its subgroups (GS 2013)

Every countable group has only countably many distinct subgroups. The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have ...
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2answers
58 views

Probability of result

Morning, A inbound contact centre says they will answer calls within 3 mins 70% of the time, if I ring the call centre 5 times over a week what is the probability I get my calls answered in under 3 ...
2
votes
1answer
178 views

Existence and uniqueness of limit of inverse function

Let $f:(a,b) \rightarrow \mathbb{R}$ be a one to one function. If $x_0$ is a point of the open interval $(a,b)$ such that $\lim_{x \rightarrow x_0} f(x) = l$, is it necessary that $\lim_{x \rightarrow ...
2
votes
2answers
50 views

Continuous bijection whose inverse is not continuous at uncountably many points

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible ...
1
vote
1answer
29 views

Let $f\geq 1$, Is the function $p\rightarrow \int |f|^p d\mu$ continuous

Let $f:X\rightarrow [0,\infty[$ be a measurable function that is greater than or equal to $1$ for every $x\in X$ and $\mu$ be a positive measure on $X$. Consider the function $g:]0,\infty[\rightarrow ...
25
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4answers
609 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
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1answer
152 views

Applications of the Pontryagin product for abelian groups

For an abelian group $G$, one can give an explicit description of the homology ring $H_*(G, k)$ for e.g. $k=\mathbb{Q}, \mathbb{Z}_p$ or in general PIDs $k$ in which every natural number is ...
16
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1answer
339 views

Examples of universal constructions in probability theory

I am looking for more examples of universal constructions in probability theory. Like the construction a of Gaussian space from a real Hilbert space, or a Poisson jump process from a measurable space ...
1
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1answer
48 views

Closed sets, boundary, topology.

Let A be a closed subset of the real numbers. It is always possible to find a subset B of the real numbers such that A is equal to the boundary of B? Prove if true, find a counterexample if not. I ...
2
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1answer
42 views

Submonoids of $\mathbb{N}^k$

Do you know if all submonoids of $\mathbb{N}^k$ are finitely generated? If not, can you give me a counter-example? EDIT : I mean $\mathbb{N}^k$ as a submonoid of $(\mathbb{Z}^k,+)$. I already know ...
4
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1answer
81 views

Does “locally connected and path-connected” imply locally path-connected?

Some friends and I discovered this question when we were studying for an exam and were trying to find examples for all combinations of topological properties we had seen in the course so far. One we ...
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2answers
32 views

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$ Where $\Omega = \{(x,y,z) : z \leq 1\}$ and $\rm{Fr}(\Omega)$ means the $\Omega$'s boundary.