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)

2
votes
1answer
121 views

Where is the fallacy in this coupling argument of two Bernoulli variables?

With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function: $$\begin{align*} f(\mathcal{J}) = ...
7
votes
3answers
568 views

Banach Algebra counterexample

Can someone give me an exemple of a Banach Algebra $\mathbb{A}$, for which there is no isometric representation in a Hilbert Space ? (with proof or references to proof) Thank you very much :)
21
votes
2answers
1k views

Is there a function with infinite integral on every interval?

Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
3
votes
2answers
167 views

Sequence convergence and parentheses insertion

find an example for a series $a_{n}$ that satisfies the following: $a_{n}\xrightarrow[n\to\infty]{}0$ ${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ does not converges There is a way to insert ...
4
votes
3answers
2k views
3
votes
2answers
135 views

Constructing a separable space that is not hereditarily separable.

The construction that I had in mind was: Let $X$ be an uncountable space. We'll assume it has just one countable dense subset $E$. Let points $p_{1},p_{2}\in X\setminus E$ such that every open set ...
1
vote
1answer
68 views

Is every perfectly normal space submetrizable?

In the previous question, Some counterexamples are given which shows shat not every perfectly normal space is paracompact. Thanks Henno. I have another question. It may be difficult: Is every ...
1
vote
1answer
151 views

Is a perfectly normal space always a paracompact space?

Is a perfectly normal space always a paracompact space? A topological space $X$ is called a perfectly normal space if $X$ is a normal space and every closed subset of $X$ is a $G_\delta$-set. ...
-2
votes
2answers
232 views

Surjective and continuous map between Hausdorff spaces

Can we say that a surjective and continuous map $p:X\to Y$ is a quotient map iff both $X$ and $Y$ are Hausdorff? If not, could you give me an example? I am terrible at finding counterexamples.
1
vote
1answer
54 views

Example ideal of $\mathfrak{sl}(2,\mathbb{C})$

I need an example about ideal from lie algebra $\mathfrak{sl}(2,\mathbb{C})$ except trivial ideal and $\mathfrak{sl}(2,\mathbb{C})$ itself, can someone help me? I try to make ideal except trivial ...
3
votes
1answer
147 views

On groups whose center has odd order

Let $G$ be a finite group such that $Z(G)$ is of odd order and $Inn(G)$ is of even order. Then prove $G\simeq Z(G)\times N$, such that $N$ is a subgroup of $G$ where $N\simeq Inn(G)$. Thank you!
1
vote
2answers
3k views

Real world well formulated examples of non linear optimization problems

I'm trying to find around the web some real world examples of non linear optimization problems. I currently need examples of: Non restringed optimziation ( $\max$/$\min$ $f(x)$ for ...
6
votes
2answers
167 views

Infinitely many zeros of a nonconstant continuous function?

Let $f:[0,1]\to\mathbb{R}$ be a nonconstant continuous function. Is $S=\{x: f(x)=0\}$ finite? I have thought of a function with countably many $0$'s like lots of triangular bumps at each point ...
1
vote
1answer
183 views

An example of Kirchhoff equation

I'm studying about a simply kind of Kirchhoff equation in one-dimensional that means $$\begin{cases} ...
8
votes
2answers
755 views

Continuity of the inverse $f^{-1}$ at $f(x)$ when $f$ is bijective and continuous at $x$.

Prove or disprove: Let $f:\mathbb{R}\to\mathbb{R}$ be bijective and $f$ is continuous at $x$. Then $f^{-1}$ is continuous at $f(x)$. Any hints are welcome. If this is false, I would like to have ...
1
vote
2answers
81 views

prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ when $n$ is odd

let $n$ be odd integer , prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ it's an example which the text proves ! but i can't understand any thing from the argument ! but i tried to ...
0
votes
1answer
167 views

Category-theoretic cross product and set-theoretic cross product

I recently proved as an exercise the associativity of cross product as defined in category theory. But in set theory, cross product is not associative. It seems intuitive to me that cross should be ...
10
votes
3answers
135 views

Is this kind of space metrizable?

It has a nice result from Tkachuk V V. Spaces that are projective with respect to classes of mappings[J]. Trans. Moscow Math. Soc, 1988, 50: 139-156. If the closure of every discrete subset of a ...
4
votes
1answer
111 views

A question on metrizable space

This exercise is from "General Topology" by Engelking: Give an example of a metrizable space which cannot be embedded in a locally compact metrizable space. I don't how to start. The ...
2
votes
1answer
68 views

Understanding an example of M. L. Wage, W. G. Fleissner, and G. M. Reed

In this paper of M. L. Wage, W. G. Fleissner, and G. M. Reed, the authors claimed that having a zeroset diagonal does not guarantee submetrizable by showing Example 2. However, the example is very ...
3
votes
4answers
125 views

A domain with only a (non-zero) prime ideal

What is an example of a domain $A$ such that Spec$A=\{(0),\mathfrak p\}$? For instance one could find a principal ideal domain that is also a local ring but I can't imagine such a ring.
2
votes
1answer
343 views

$L^p$ integrable but not $L^q$ integrable

Does there exist a continuous function on $[0, \infty)$ such that it is in $L^p(0,\infty)$ for some $p\in [1,2]$ but is not in $L^q(0,\infty)$ for any $q\in (2, 2/(2-p))$? Thanks!
2
votes
1answer
586 views

An example of a generalized Cantor set with positive Lebesgue measure [duplicate]

I want to know if there exist a set $ X\subset \mathbb R$ such that $X$ is $i)$ Perfect $ii)$ Compact $iii)$ Has empty interior $iv)$ Totally disconnected $v)$ Is not countable But $X$ has ...
7
votes
3answers
865 views

Examples of perfect sets.

Let $0\lt a\lt 1$. Can I get examples of of subsets of $[0,1]$ that are perfect sets, contains no intervals and has measure $1-a$. Well, I know by construction the Cantor set is perfect, contains ...
7
votes
2answers
161 views

What are the possible minimal acl-dimensions of strongly minimal models?

The question is as in title. By acl-dimension I understand the cardinality of maximal acl-independent set (well-defined for strongly minimal theories). By minimal I understand that there is no ...
32
votes
1answer
2k views

An example of a division ring $D$ that is **not** isomorphic to its opposite ring

I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring $D$ that is not isomorphic to its opposite ring. ...
1
vote
2answers
295 views

Weierstrass M-test proof?

Let (X,d) be a metric space. For each n $\epsilon$ N let $g_n$:X$\rightarrow$R be a continuous function. Let ($a_n$) be a sequence of positive real numbers such that the series $\sum_n_=_1^\infty a_n$ ...
1
vote
2answers
316 views

Counterexample to inverse Leibniz alternating series test

The alternating series test is a sufficient condition for the convergence of a numerical series. I am searching for a counterexample for its inverse: i.e. a series (alternating, of course) which ...
1
vote
0answers
98 views

Semisimple commutative Banach algebra

On a semisimple commutative Banach algebra all Banach algebra norms are equivalent. Is this true without assuming semisimplicity?
1
vote
1answer
44 views

Real commutative Banach algebra with identity

I am looking for example of real commutative Banach algebra with identity which does not admit a nonzero real multiplicative linear functional
3
votes
2answers
177 views

finding a function with predetermined $f^{(n)}$s at $0$ and $1$.

Let $a_n,b_n$ be 2 arbitrary sequences of real numbers. Is there any $C^\infty$ function $f:\mathbb{R}\to\mathbb{R}$ such that for any $n$: $$f^{(n)}(0)=a_n,\quad f^{(n)}(1)=b_n$$
2
votes
1answer
71 views

For every monoid $M$ with zero is there a group $G$ such that $\mathrm{Grp}(G,G)\cong M$?

All monoids that I will consider here have identities. A monoid $M$ is said to have a zero iff $\exists z\in M \forall x\in M (zx=xz=z)$. Let $M$ be any monoid with a zero. Must there exist a group ...
-1
votes
2answers
106 views

If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant? [closed]

If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?
0
votes
1answer
127 views

Strict coisometries and operator norm.

I got stuck at the following problem. Let $X,Y$ be normed spaces. A bounded linear operator $\tau\in\mathcal{B}(X,Y)$ is called strictly coisometric if $$ ...
7
votes
2answers
95 views

Regular $T_2$ space which is not completely regular.

Theorem 10. of Pontryagin's Topological Groups says that: Every Hausdorff topological group is completely regular. But is there exists a Regular $T_2$ space which is not completely regular?
10
votes
3answers
968 views

Is every countable space first countable?

All of the examples of non-first countable spaces I have seen are uncountable (for instance any uncountable set with the cofinite topology). I would like to know if every countably infinite $T_1$ ...
0
votes
1answer
48 views

Sufficient condition for reducibility of polynomial $f(x,y)$

[Dual to this question] Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied: 1) For every $x_0\in\mathbb{Q}$, the polynomial ...
2
votes
1answer
73 views

Looking for example of an order homomorphism that doesn't preserve joins.

I know that not every order homomorphism preserves joins. But, I can't think of an example! Both minimal examples and 'natural' examples welcome.
2
votes
1answer
102 views

Sufficient condition for irreducibility of polynomial $f(x,y)$

Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied: 1) For every $x_0\in\mathbb{Q}$, the polynomial $f(x_0, y)\in\mathbb{Q}[y]$ ...
7
votes
2answers
696 views

True/False Questions for Complex Analysis

I am studying for my (introductory) complex analysis final exam tomorrow. I am practicing an old final exam, which unfortunately has no answer key. Here is a link: ...
5
votes
0answers
109 views

An interesting space inspired by Mrowka's space

The example is from Bell M G. First countable pseudocompactifications[J]. Topology and its Applications, 1985, 21(2): 159-166.. Let us recall some necessary definitions firstly: Let $X$ be a ...
4
votes
1answer
199 views

Explicit counter-example to corona problem

The corona problem is known to fail for the complex polydisk, for dimension greater than 2. Does anyone has an explicit example of such functions?
1
vote
1answer
247 views

How to directly show that Figure 8 injective immersion is not a monomorphism

I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an ...
1
vote
2answers
166 views

Is there a residually finite group not finitely presented?

I am looking for a residually finite group which is not finitely presented. Does such a group exist?
4
votes
0answers
79 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
3
votes
1answer
318 views

A completely regular topological space which is $T_0$ but not $T_1$?

The question pretty much says it all. I need a completely regular (the definition not requiring $T_1$) topological space which is $T_0$ but not $T_1$. I've sifted through Counterexamples in Topology ...
0
votes
1answer
157 views

Unique nearest point in epsilon neighborhood of compact real manifold?

I have to proof the following assertion: Let $X$ be a compact submanifold of $\mathbb{R}^n$ and $\mathcal{U}^\varepsilon=\{p\in\mathbb{R}^n\;:\; |p-q|<\varepsilon \text{ for some }q\in X\}$. Then ...
2
votes
2answers
200 views

example of homotopy which is not path homotopy

Can someone give me a simple, concrete example of a homotopy, which is not a path homotopy? Let $f, f'$ be continuous maps from $X$ to $Y$, and let $F: X \times I\to Y$ a continuous map such ...
3
votes
2answers
107 views

Metric spaces and distance functions.

I need to provide an example of a space of points X and a distance function d, such that the following properties hold: X has a countable dense subset X is uncountably infinite and has only one ...
3
votes
2answers
295 views

Making Tychonoff Corkscrew in Counterexamples in Topology rigorous?

I'm reading pages 109 and 110 of Seebach and Steen's Counterexamples in Topology (p. 61 here) and I don't understand one of their steps. In particular, at the bottom of page 109 they say, "by ...