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 ...

learn more… | top users | synonyms (2)

17
votes
9answers
454 views

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
1
vote
3answers
57 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
1
vote
0answers
16 views

Complete lattice mod a congruence relation.

Suppose $(X,\le)$ is a lattice and $R$ is a congruence relation on it. If $X$ complete, is $\frac{X}{R}$ complete?
1
vote
1answer
46 views

non-atomic complete Boolean lattice

Is there a Boolean complete lattice that is not atomic?
2
votes
1answer
63 views

Intuitively confusing example of open set and topology on Real Line

My question occurs when I see the problem that show that every continuous function is borel measurable. I know that for topological space $(X,\tau)$, we define that "open sets" means sets in topology. ...
4
votes
0answers
108 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
1
vote
2answers
26 views

Question with real function and connectedness

I asked myself: If $f: \mathbb R \to \mathbb R$ and $f(\mathbb R)$ is connected then does $f$ map connected sets to connected sets? My idea is that is true: There exists $f: \mathbb R \to \mathbb R$ ...
6
votes
1answer
123 views

Counterexample to “Measurable in each variable separately implies measurable”

Some fellow classmates are preparing for a qualifying exam on real analysis, and asked me for help on the following question: Let $ \ f:[0,1]^2\longrightarrow\mathbb{R}$ be such that: (i) $\ ...
0
votes
1answer
21 views

Order isomorphic but not Boolean isomorphic

Is there an example of Boolean lattices $(X,\le)$ and $(Y,\le)$ and a function $$f:X\to Y$$ such that for all $x,z\in X$ $$x\le z ~~~~ \leftrightarrow ~~~~f(x)\le f(z) $$ so that for some $a\in X$, ...
0
votes
0answers
26 views

A set with a property in a Boolean lattice.

Are there a Boolean lattice $(X,\le)$, $A\subseteq X$ and $b\in X$, such that $\sup A$ exists but $\sup\{a\wedge b|a\in A\}$ does not exist.
0
votes
1answer
28 views

Simple question about montonically increasing function

Suppose we have a continuous $f: \mathbb{R} \to \mathbb{R}$, we know the definition of a monotonically increasing function is for $x,y \in \mathbb{R}$, if $x < y$ then $f(x) < f(y)$. I know that ...
7
votes
2answers
152 views

Mutually densely-embedded non-homeomorphic topological spaces

My question is the following: Do there exist two non-homeomorphic topological spaces $X$ and $Y$ such that there are embeddings $f : X \hookrightarrow Y$, $g : Y \hookrightarrow X$, with both ...
162
votes
30answers
11k 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 ...
2
votes
1answer
95 views

non-prime maximal ideal in a complemented lattice

Is there a complemented lattice, which has a non-prime maximal ideal?
2
votes
3answers
68 views

An example of a non first countable Fréchet-Urysohn space?

As the head title says, I need a Fréchet-Urysohn space but not first countable, (on the way, a good Text book to follow). Thanks.
4
votes
1answer
48 views

Topological space in which there are no close and compacts subsets (except for the empty set)

Any example of those topological spaces? I cant think of no one :S I think it must be infinite and it must not be T2, but no idea how to find one.
1
vote
3answers
96 views

About the Pigeonhole principle

The principle says that: Let $k$ and $n$ be any two positive integers. If at least $kn+1$ objects are distributed among $n$ boxes, then one of the boxes must containat least $k+1$ objects. In ...
2
votes
4answers
107 views

Boolean prime ideal theorem and the axiom of choice

The Boolean prime ideal theorem is strictly stronger than ZF, and strictly weaker than ZFC. I'm looking for nice examples (like the existence of non-measurable set) that request at least that theorem ...
1
vote
0answers
40 views

Resolvable spaces

a space $X$ is called a resolvable space if it is expressible as a union of two disjoint dense subsets. I want to find a resolvable but not lindelof space? Is there any example such a space?
8
votes
4answers
2k views

Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
8
votes
2answers
469 views

What is the intuition between 1-cocycles (group cohomology)?

This is, I'm sure, an incredibly naive question, but: is there a simple explanation for why one should be interested in 1-cocycles? Let me explain a bit. Given an action of a group $G$ on another ...
9
votes
1answer
105 views

Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?

The particular point topology on any set is connected, but on removing the particular point, the complement is discrete, and hence totally disconnected. Although this is not even $T^1$, Cantor's leaky ...
2
votes
0answers
46 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
2
votes
3answers
168 views

Counterexample to linear transformation.

What counterexample can I use to prove that ($ \mathbb{R}_{[x]}$is any polynomial): $L :\mathbb{R}_{[x]}\rightarrow\mathbb{R}_{[x]},(L(p))(x)=p(x)p'(x)$ is not linear transformation. I have already ...
2
votes
2answers
52 views

K topology: Examples

Why would the interval $(-3,1)$ be open in the $k$-topology? (I'm using Munkres). Can I have some other examples of intervals in $k$-topology? What exactly does $(a,b)$ $\cup$ $(a,b)-k$ for ...
0
votes
1answer
47 views

Analouge of the Mean value theorem for holomorphic functions

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be entire. Let $w_1,w_2$ be any two distinct complex numbers. Must there exist $c\in \overline{B_{|w_2-w_1|}(w_1)}$ such that ...
3
votes
1answer
77 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
1
vote
0answers
18 views

Is a extension of a premeasure preserves outer-measure generated by the premeasure?

I have proved the follow: Let $X$ be a set. Let $S$ be a semi-ring of subsets of $X$. Let $\mu$ be a premeasure on $S$. Let $\overline{\mu}$ be a premeasure on a ring generated by ...
4
votes
1answer
118 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?
10
votes
1answer
239 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
2
votes
2answers
40 views

Class of algebras defined by quasi-identities is not closed under taking the quotients

Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By quasi-identity I mean the formula of the form $$(\forall x_1) (\forall x_2) \dots (\forall x_n) ...
3
votes
3answers
138 views

What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
2
votes
0answers
25 views

A question on the classical Mrowka space

Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset ...
0
votes
2answers
100 views

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$? [closed]

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$ ? Im sorry I am confused about ring theory. Its all new to me.
11
votes
1answer
175 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
4
votes
2answers
50 views

Examples of ergodic geodesic flow

Are there any good examples of a geodesic flow that is ergodic? I know the result that states that the geodesic flow for manifolds with negative curvature are ergodic, but I'm fishing for some ...
24
votes
3answers
487 views

Is every function with the intermediate value property a derivative?

As it is well known every continuous function has the intermediate value property, but even some discontinuous functions like $$f(x)=\left\{ \begin{array}{cl} \sin\left(\frac{1}{x}\right) & x\neq ...
8
votes
2answers
141 views

Counterexample: multiplying modules by elements of an ideal vs. taking linear combinations

Let $R$ be a ring (commutative, unital) and $M$ an $R$-module. Let $I \subset R$ be an ideal. We make the following definitions: $$ A := \{ am \ | \ a \in I,\ m \in M \} $$ $$ B := \left\{ ...
4
votes
1answer
73 views

The function $f(x)=(x\vee a)\wedge b$ in a lattice.

Is there an algebraic modular lattice $(X,\vee,\wedge)$ and $a,b\in X$ with $a\le b$ such that the function $$f:X\to X$$ $$f(x)=(x\vee a)\wedge b$$ is not $\vee$-homomorphism?
6
votes
1answer
165 views

Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?

This is a follow-up to Continuous partials at a point but not differentiable there?, but I'll make this question self-contained. Throughout, $f$ will denote a function $\mathbb{R}^2\to\mathbb{R}$. An ...
2
votes
2answers
195 views

Non-measurable set in product $\sigma$-algebra s.t. every section is measurable.

Let $\Omega$ and $\Gamma$ be two nonempty sets and $\mathscr{A}$ and $\mathscr{B}$ be $\sigma$-algebras over $\Omega$ and $\Gamma$, respectively. The product $\sigma$-algebra of $\mathscr{A}$ and ...
1
vote
1answer
68 views

Is every Hilbert space an $L^2$ space

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
0
votes
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?
0
votes
0answers
57 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
votes
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
votes
1answer
99 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
votes
2answers
645 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 ...
1
vote
1answer
46 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 $\epsilon >0$ ...
1
vote
1answer
42 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 ...
0
votes
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.