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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Discontinuous maps taking compacts to compacts

It's commonly known that in general topology, a continuous map $f$ from a topological space $(X, \tau)$ to another topological space $(Y, \tau')$ will send every compact set to another compact set. ...
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1answer
29 views

Counterexample to a variation on “The politician theorem”.

The following is a theorem in graph theory that has a nice 'real world' interpretation: Suppose $G$ is a finite simple graph in which any two vertices have precisely one common neighbour. Then ...
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2answers
457 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 ...
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32answers
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What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ...
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857 views

A locally injective but non globally injective function?

A continuous function $f : U \subset \mathbb{R}^n \to \mathbb{R}^n$, is said to be locally injective at $x_0 \in U$ if exist a neighborhood $V \subset U$ of $x_0$ s.t. $f|_V$ is injective. $f$ is said ...
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1answer
65 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 $\alpha \in V$ ...
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2answers
255 views

Is projection of a closed set $F\subseteq X\times Y$ always closed?

If we have closed subset $F$ of product $X \times Y$ (product topology) does it mean that $p_1(F)$ (projection on first coordinate) is closed in $X$ and $p_2(F)$ in $Y$ are closed? If not, why not ...
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53 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:
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2answers
358 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 ...
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2answers
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find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges

Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are ...
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1answer
79 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} ...
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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, $f(S)\subset Y$ ...
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3answers
125 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 .
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2answers
48 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
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0answers
71 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 $[0,\infty)$. ...
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1answer
28 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 ...
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1answer
192 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
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27 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 ...
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1answer
23 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 ...
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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 ...
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1answer
33 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 ...
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2answers
93 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 ...
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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) ...
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2answers
100 views

Can an algebraic structure such that $x+a=x+b$, have solutions for all $a,b∈\mathbb{K}$ exist?

Does there exist an algebraic structure $(\mathbb{K},+)$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b\in \mathbb{K}$?
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1answer
42 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 ...
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4answers
736 views

Example for a proper dense subspace?

I have been reading some books on functional analysis, and many of them keep talking about a vector space along with a dense proper subspace of it (especially when constructing counterexamples). But ...
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1answer
45 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 ...
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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 ...
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2answers
41 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 ...
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3answers
47 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 ...
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1answer
250 views

convergence of sequence of averages the other way arround

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \dfrac{x_1 + \cdots+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \dfrac{x_1 + ...
4
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1answer
70 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 $f'$ ...
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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 ...
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0answers
62 views

Counter-example to exponential law for locally compact [non-Hausdorff] spaces

There is a natural bijection $$ \operatorname{Map}(X\times Y,Z)\cong\operatorname{Map}(X,\operatorname{Map}(Y,Z)),\quad f\mapsto(x\mapsto(f(x,{-})). $$ If $X$, $Y$, $Z$ are topological spaces one can ...
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0answers
52 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 ...
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3answers
33 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.
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1answer
44 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: ...
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0answers
29 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 ...
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3answers
255 views

Is every monoid isomorphic to its opposite

This may be a trivial question. Every group is isomorphic to its opposite using the isomorphism that sends $x$ to $x^{-1}$. Now does this hold even if the condition of existence of inverses is ...
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0answers
24 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 ...
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3answers
161 views

A function not differentiable exactly two points of $[0,1]$. construction of such a function is possible?

Can a continuous function on $[0,1]$ be constructed which is not differentiable exactly at two points on $[0,1]$ ?
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1answer
51 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 ...
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1answer
29 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 ...
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336 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 ...
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68 views

All possible total orderings of a finite set are isomorphic. What are some other examples of this phenomenon?

All possible total orderings of a finite set are isomorphic. I find these kinds of results remarkable. Here's a few more. Assume that $S$ is a finite set. Then: All possible field structures on $S$ ...
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1answer
34 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 ...
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0answers
41 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}, ...
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6answers
473 views

Is $A^2=I \implies A=\pm I$ necessarily true?

$A$ is $n\times n$ matrix. How to prove whether it is true or false $$A^2=I \implies A=\pm I$$ I was trying on $2\times 2$ case...multiplying general entries and then equating them to the identity ...
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1answer
342 views

An equivalent definition of a group

Exercise 15 from Hungerford: Algebra. Let $G$ be a nonempty finite set with an associative binary operation such that for all $a,b,c\in G\,\,ab=ac \Rightarrow b=c$ and $ba=ca \Rightarrow b=c$. Then ...
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1answer
309 views

If product $X_1\times X_2$ and $Y_1\times Y_2$ are homeomorphic is $X_i\simeq Y_i$?

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...