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)

0
votes
0answers
30 views
+50

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
2
votes
0answers
48 views

Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
5
votes
2answers
350 views

An example for infinite dimensional vector space with Hamel dimension smaller than $\operatorname{card} F$

What will be the example for a vector space(infinite dimensional) over a field where Hamel basis has strictly smaller cardinality than that of field? It is not possible in a Hilbert Space (over R or ...
3
votes
0answers
36 views

Reconciling two different definitions of constructible sets

This question is really about sets and topology, but it is motivated from commutative algebra, hence the tag. Setup: Let $X$ be a set and let $\{U_\lambda\}_{\lambda\in\Lambda}\subset 2^X$ be a ...
4
votes
1answer
64 views

Non-example for vector space

$V$ is a vector space over a field $F$ if it satisfies the following conditions. $(V,+)$ is an abelian group. $1 \in F $ such that $1.\alpha=\alpha$ for every $\alpha \in V$ ...
3
votes
1answer
76 views

Counter exchanging limit and integral

Background I came across this answer on Math SE which claimed it made a lot of sense to switch limit and integral. In response I came up with the following counter-examples: $\lim_{w \to 0} ...
2
votes
3answers
81 views

Looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points

I am looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points ; please help , thanks in advance .
1
vote
2answers
46 views

If $\sum a_{2k}$ exists then $\sum a_m$ exists?

Let $\{a_0,a_1,a_2...\}$ be a sequence of real numbers let $s_n=\sum a_{2k}$. If $\lim_{n\rightarrow \infty} s_n $ exists then $\sum a_m$ exists. Is it true? I don§t find a counter example
0
votes
1answer
26 views

If the $(n-1)$th moment exists does the $n$th moment necessarily exist?

Let's suppose the distribution is unknown but that the second moment is known to be finite. Doesn't this imply that the distribution should fall off exponentially fast and therefore higher moments ...
2
votes
0answers
20 views

Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
0
votes
1answer
21 views

A simple example of a base point of a linear series

I'm reading Fulton's algebraic curves book and he make the following definition of linear series (page 110): Let $D$ be a divisor, and let $V$ be a subspace of $L(D)$ (as a vector space). The ...
1
vote
1answer
31 views

What's an example where this definition of measure has a limit that does not exist?

I'm looking for an example. Consider defining the measure of a set $E \subseteq \Bbb{R}^d$ by a limit: Take $$m(E) := \lim_{N \to \infty} \frac{1}{N^d} \cdot \left| E \cap \frac{1}{N} \Bbb{Z}^d ...
2
votes
1answer
27 views

Convergence in distribution with finite mean

I'm preparing myself for the final exam of my graduate Probability Theory course and was stuck with another one of the exercises our professor gave us. Let $X_n, n=1,2,\ldots,$ and $X$ be nonnegative ...
3
votes
2answers
90 views

Does there exist a sequence $\{a_{n}\}$, but $\lim_{n\to\infty}a_{n}\neq 0$

Does there exist a sequence $\{a_{n}\}$, such that $$\lim_{n\to\infty}(a_{n+1}-a_{n})=0\ \ \ \text{and} \ \ \lim_{n\to\infty}\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}=0$$ but $$\lim_{n\to\infty}a_{n}\neq 0 ...
0
votes
0answers
16 views

example of a quadratic form

Would someone be able to tell me an example of a quadratic form defined on a five-dimensional vector space $V$ over a non-archimedean local field $k$ of positive characteristic (not equal to two, say) ...
2
votes
1answer
31 views

Counterexample to Rellich-Kondrachov Compactness Theorem, case $q=p^*$

I was searching some counterexample for I was searching some counterexample for Rellich-Kondrachov Compactness Theorem (You can see: PDE, Evans, chapter 5), for the case $q=p^*$ and I found this ...
2
votes
0answers
70 views

Finding a weight function with a specific property

I am looking for a (smooth, quickly decaying) function $w : [0,\infty) \rightarrow \mathbb{R}$ such that $$w(t) \cdot \int_{0}^{t} \frac{1}{w(2x)} dx $$ is absolutely integrable on $[0,\infty)$. ...
5
votes
1answer
44 views

Is there a nowhere differentiable norm on $\Bbb R^n$?

Is there a nowhere differentiable norm on $\Bbb R^n$? The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable ...
0
votes
0answers
22 views

More on the implicit function theorem: is this example correct?

I am trying to understand the implicit function theorem so I thought it would be a good idea to work out an example. Please could someone look at this and tell me if it is correct? Consider the ...
0
votes
2answers
33 views

Set theory basics exam

I have a question about this, we had this on our exam. Let be $f:A \to B$ a function. Prove next statements or give an example against it. (i): if $A$ is countable, then $f(A)$ is also ...
2
votes
3answers
41 views

Interesting examples of non-normal operators?

I am currently learning spectral aspects of linear algebra. At first sight, it seems like normality is very narrow restriction. But, I can not think up any examples of non-normal operators. There is ...
4
votes
1answer
64 views

Counter Example of Continuous Functions

I came across this question: Question: Let $f$ be a real continuous function defined on [0,1] such that $f(0) = 0$ and $f(1) = 1$. Prove or give a counter-example to the following: a) If $f'$ ...
4
votes
2answers
22 views

Non-algebraic subfield intersection

Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. I'm having trouble thinking of an explicit example ...
0
votes
3answers
25 views

A semigroup with three or four elements without identity [duplicate]

Does there exist a semigroup with three or four (or finite) elements, without identity? I tried to construct such an example, but every example I tried to construct had an identity element.
4
votes
1answer
39 views

Anomalous finitary objects

I think it was about 15 years ago that Igor Pak told me that the symmetric group on six elements has outer automorphisms, and I was startled. Somehow that had escaped my notice. That one is a freak: ...
2
votes
0answers
23 views

Monomorphisms into direct products

Let $G$ be a group. I am interested in the following property: For any groups $A,B$ and monomorphism $G \hookrightarrow A \times B$, either $G \hookrightarrow A$ or $G \hookrightarrow B$. For ...
0
votes
1answer
48 views

Differentiability at a point does not imply differentiability on a neighbourhood. Do all similar implications fail?

(Write $[]$ for the Iverson bracket.) I recently learned about the function $$\mathbb{R} \rightarrow \mathbb{R}$$ $$x \mapsto [x \in \mathbb{Q}]x^2 +[x \notin \mathbb{Q}](-x^2)$$ which is ...
0
votes
1answer
25 views

Vector field along a map

Let $U \subset \mathbb R^n$ be open and $f: U \to \mathbb R^m$ be smooth. Consider the vector fields ${\partial f \over \partial x_i}$. I think these are an example of a vector field along $f$ but ...
0
votes
1answer
33 views

Show that cauchy sequences are not topological terms, find a counterexample?

I need to show that the expression 'is a cauchy sequence' is not a topological expression. I could show this by finding a homeomorphic function $\phi$ such that a Cauchy sequence is mapped onto a ...
0
votes
0answers
39 views

Give an example of a locally compact Hausdorff space which is not compact.

Give an example of a locally compact Hausdorff space which is not compact. Would the following be valid? Could someone give an example using a different topology? $(\mathbb{R}, ...
7
votes
3answers
319 views

Space on which all real-valued continuous functions achieve maximum but not compact?

A friend is writing a book for non-mathematicians; he has asked me some questions... One possible direction I suggested was whether a topological space (metric space can probably be assumed given what ...
1
vote
0answers
47 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
3
votes
2answers
56 views

In a metric space $(X, d)$, if closed sets $A$, $B$ contain sequences $a_n,b_n$ such that $d(a_n,b_n)\to 0$, must $A\cap B\neq \emptyset$?

I wrote a test yesterday in which one of the questions asked us to prove that if $A$ and $B$ are disjoint closed subsets of a metric space $(X, d)$, then there exist disjoint open subsets $U$ and $V$ ...
12
votes
2answers
296 views

Topology on $\mathbb{R}$ strictly coarser (resp. finer) than the usual one which is still Hausdorff (resp. connected)

The following are simple observations. Suppose $\mathcal{T}_1,\mathcal{T}_2$ are two topologies on a set $X$ such that $\mathcal{T}_1$ is finer than $\mathcal{T}_2$. If $( X ,\mathcal{T}_2 ...
0
votes
4answers
77 views

Counterexample for $\lim \limits_{x \to c} \left\lvert f(x) \right\rvert = \left\lvert L \right\rvert$ then $\lim \limits_{x \to c} f(x) = L$ [duplicate]

I know that the converse of this is true. I was looking for some counterexamples proving this statement false.
2
votes
2answers
45 views

Give an example of a topological space $X$, and a connected component which is not open in $X$

Give an example of a topological space $X$, and a connected component which is not open in $X$ I know of the following theorems: Each connected component is closed; If a topological space has a ...
4
votes
1answer
53 views

Is there a finite abelian group $G$ such that $\textrm{Aut}(G)$ is abelian but $G$ is not cyclic?

Is there an example in which $G$ is a finite abelian group and $\textrm{Aut}(G)$ is abelian but $G$ not cyclic?
1
vote
0answers
25 views

Linear system without base points

Let $D$ be a divisor and $V$ a vectorial subspace of $L(D)$. The set of the divisors $S=\{(f)+D\mid f\in V\}$ is called a linear system. A point $P\in C$ is a base point if for every divisor $D\in S$, ...
4
votes
2answers
207 views

Space which is neither locally connected at any point nor totally disconnected

Let $X$ be a topological space; then we say that $X$ is locally connected at $x$ if $x$ admits a neighborhood basis of open, connected sets. In this sense, a space is locally connected iff it is ...
34
votes
3answers
1k views

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
1
vote
1answer
22 views

Can a field extension still have “non-separability” above its maximal purely inseparable subextension?

Question 1 Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely ...
2
votes
0answers
47 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
2
votes
1answer
26 views

Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
1
vote
0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
5
votes
3answers
34 views

Possible textbook redundancy concerning invertible mappings

In my textbook (Modern Algebra by John Durbin, 6th Ed), there is the following theorem: Let $S$ denote any nonempty set. (a) Composition is an associative operation on $M(S)$, with identity ...
1
vote
0answers
28 views

core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
8
votes
8answers
265 views

An example of a mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that $\eta(x)=n$ has infinitely many solutions for each $n\in\mathbb{N}$

Suppose I have the mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\eta(x)=n$ has infinitely many solutions. I saw this question which is basically the ...
4
votes
1answer
71 views

Is it true that the intersection of a sequence $K_1 \supset K_2 \supset K_3 \dotsm$ of connected subsets of $\mathbb{R}^2$ is also connected?

I have got one counterexample for this : Consider the family {D} of closed discs centered at zero having radius $1+1/n$, i.e. disc $D_1$ has radius $1+1=2$, $D_2$ has radius $1+1/2=1.5$, and so on. ...
1
vote
0answers
32 views

Is imaged of a Polish Hausdorff space under an injecitve map always Hausdorff?

I have a question about Hausdorff topological space. Question: Let $X,Y$ be topological spaces. If $X$ is a Polish space (i.e. $X$ is a separable and completely metrizable space.) and $Y=f(X)$ ...
2
votes
2answers
24 views

Does one of these conditions on a sequence imply the other one?

Let ${(r_n)}_{n \geq 0}$ be a sequence of integers $\geq 2$. Set $q_n=\prod_{i=0}^{n-1} r_i$ (agreeing with $q_0=1$). I want to know whether one of these two conditions implies the other one (I think ...