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
27 views

Counterexample to $G_2/G_1$ abelian implies $(A\cap G_2)/(A\cap G_1)$ abelian.

My officemate and I are currently procrastinating on research/grading (by doing some math, obviously), and he came up with this statement, which we are both convinced is false, but are having trouble ...
1
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1answer
8 views

Image of a product of opens

This is a general topology question. Let $k < n$ be positive integers. Suppose we have opens $U \subset \mathbf R^k$ and $V \subset \mathbf R^{n-k}$ and a continuous and injective map $$f: U ...
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3answers
71 views

Does A∪C=B∪C imply A = B? What about intersection instead of union?

if A, B and C are sets; for (u = union, n = intersection) $A \cup C = B \cup C$, would $A = B$ ? Similar for $A \cap C = B \cap C$, would $A = B$ ? For my conclusion I have got for the ...
2
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0answers
24 views

Linear functional on Schwartz space non-continuous

I'm looking for a linear functional on the Schwartz space $\mathcal S(\mathbb R^n)$ which is not a tempered distribution. I appreciate any help :)
2
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2answers
83 views

change the order of sup and inf

I know that for a series $a_{m,n}\ge 0$ $$\sup_m \inf_n a_{m,n}\neq \inf_n\sup_m a_{m,n}$$ But I can't find a counterexample. Could anyone help me? Thanks a lot!
5
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1answer
52 views

Induced measure needs not to be sigma-finite

Let $(\Omega, \mathcal{F}_1, \mu)$ be a $\sigma$-finite measure space, Let T:$\Omega \to \mathbb R$ be $\langle\mathcal{F}, B(R)\rangle$-measurable. Could anyone think about an example to show $\mu$'s ...
1
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1answer
43 views
1
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35 views

Convergence in $L_p$ does not imply “a.s.” convergence

I'm having a hard time understanding some examples about this that I found on stack exchange. So I'd like to ask for the simpler possible example where a sequence of random variables $\{T_n\}$ ...
0
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0answers
31 views

How to Prove It,4.4,Ex 23, Partial Orders

Theorem Suppose $A$ is a set, $F ⊆ P (A)$, and $F \ne ∅$. Then the least upper bound of $F$ (in the subset partial order) is $∪F$ and the greatest lower bound of $F$ is $∩F$. Proof: Since any element ...
0
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1answer
51 views

Example of an additive but not linear map beetween real topological linear spaces

Let $X$ and $Y$ be real topological linear spaces. It's easy to prove that if $f\colon X \to Y$ is additive and continuous, then it's linear. I'm looking for a counterexample if the continuity ...
9
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3answers
422 views

Do subgroup and quotient group define a group?

Does a (normal) subgroup along with its corresponding quotient group define a group completely? Or are there groups with isomorphic normal subgroups and isomorphic corresponding quotient groups which ...
1
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0answers
45 views

p - norms inequality

For $v \in \mathbb{R}^n$ denote by $||v||_p$ its p-norm. That is $(\sum_{i=1}^nv_i^p)^{\frac{1}{p}}$ where $v_i$ are the componenets of $v$. I'm looking for a way to bound the following expression: ...
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1answer
64 views

Example of a function s.t. $f\in C^{1}$ and $f'\notin L^{1}$?

Let us consider in one dimension. Let $a>0$ be a given constant, $$ C[0,a]:=\{f:[0,a]\to\mathbb{R} \mid \text{$f$ is continuous in $[0,a]$}\}, $$ $$ C^{1}(0,a]:=\{f:(0,a]\to\mathbb{R} \mid ...
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2answers
59 views

non-trivial non-repetitive aperiodic tiling of the plane

Which is the less trivial example of non-repetitive aperiodic tiling of the plane you know? I cannot come up with a famous non-repetitive tiling. Are there any? A tiling is repetitive if for every ...
4
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2answers
129 views

Good example showing why limits must exist in limit product rule

I'm looking for a way to show my calc 1 students not to use the limit laws without knowing that the individual limits exists. I could use $$\lim_{x\to 0} x^{2} \sin(1/x),$$ but by doing it wrong, one ...
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2answers
60 views

Non-constant continuous function having uncountably many zeros?

I want to have an example of a non-constant function which has uncountably many zeros? Is the following function continuous? $f(x)=\prod\limits_{\alpha\in \mathbb{R\setminus Q}}(x-\alpha)$ If it so, ...
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8answers
111 views

Example where $x^2 = e$ has more than two solutions in a group [closed]

Show by means of an example that it is possible for the quadratic equation $x^2 = e$ to have more than two solutions in some group $G$ with identity $e$.
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4answers
61 views

Functions such that $f(g(x)) = x$ but $g(f(x)) \neq x $ [duplicate]

I want to find functions $f: A \to B$ and $g: B \to A$ such that $g(f) = i_A$ but $f(g) \neq i_B$. Is this possible? An exercise for one of my classes is to actually prove that if $g(f) = i_A$, ...
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2answers
22 views

When is $\Pr(U\leq v|V=v)$ not an increasing function of $v$?

Suppose $U$ and $V$ are random variables. I'm trying to find an example for which $\Pr(U\leq v|V=v)$ it not an increasing function of $v$. Current thoughts: when $U$ and $V$ are independent, ...
2
votes
3answers
39 views

Give a counterexample to $\bigcup\limits_{t \in T} (A_t \cap B_t)= \bigcup\limits_{t \in T} A_t \cap \bigcup\limits_{t \in T} B_t$

Let $\{A_t\}_{t \in T}$ and $\{B_t\}_{t \in T}$ be two non-empty indexed families of set. Find a counterexample to $$\bigcup\limits_{t \in T} (A_t \cap B_t)= \bigcup\limits_{t \in T} A_t \cap ...
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1answer
53 views

A generalized Alexandroff space that is not an Alexandroff space

A topological space $(X,T)$ is called generalized Alexandroff if any intersection of open sets is generalized open (where $A$ is generalized open if its interior contains all closed subsets of $A$). ...
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votes
2answers
68 views

Example of a function $f:[0,1] \to R$ which is differentiable on $(0,1)$ but not at the points $0$ and $1$ [closed]

Define a function $f : [0,1] \to R$ which is continuous on $[0,1]$, differentiable on $(0,1)$, and satisfies $f (0) = f(1) =0$, but is not differentiable at $x=0$ and $x=1$. Find a point $c\in(0,1)$ ...
2
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1answer
70 views

Prove or give Counterexample: Is this a Basis.

Prove or give a counterexample: If $v_1, v_2, v_3, v_4$ is a basis of $V$ and $U$ is a subspace of $V$ such that $v_1,v_2 \in U$ and $v_3 \notin U$ and $v_4 \notin U$, then $v_1,v_2$ is a basis of U. ...
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0answers
35 views

Is there a simple closed curve which is not rectifiable?

Let $\gamma:[0,1] \rightarrow \mathbb{C}$ be a loop such that $\gamma$ is injective on $(0,1]$. (So the image of the curve is a simple close curve is homeomorphic to $S^1$. Is $\gamma$ rectifiable in ...
3
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1answer
69 views

Intersection of clopen sets that contain x is the connected component of x (if X is compact)

Let $X$ be a topological space, $x\in X $, $C$ is a connected component of $x$. Define $A$ to be the intersection of all the open-and-closed sets that contain $x$ (also called the pseudo-component ...
5
votes
2answers
437 views

Bounded sequence which is not convergent, but differences of consecutive terms converge to zero

I have a question that says "Show that there is a bounded sequence $x_n$ which is not convergent but has the property that $x_n - x_{n+1} \to 0$ as $n \to 0$. What does this mean? Do I need to come ...
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0answers
21 views

The set where $f_n \to f$ is measurable? A question concerning Paul Mayer's Book

On the book Probability and Potentials of Paul Mayer page 9 one reads: Is there an example of a sequence of measurable functions taking values in a incomplete metric space such that the set where ...
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2answers
37 views

Distribution with two (or more) medians

Is there any example with a distribution with two or more medians? I was reading about the median on wikipedia: https://en.wikipedia.org/wiki/Median and here it says that there may be more than one ...
2
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1answer
88 views

$L^p$ submartingale convergence theorem

In class today, we learned about the familiar $L^p$ martingale convergence theorem: For $p >1,$ if $X_n$ is a martingale with $\sup \mathbb{E}|X_n|^p <\infty$, then $X_n \rightarrow X$ a.s. and ...
1
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1answer
89 views

Why $(\mathbb Q\times\mathbb Q)/(\mathbb Z\times{=})$ is not homeomorphic to $(\mathbb Q/\mathbb Z)\times(\mathbb Q/{=})$?

Let $\mathbb Q$ be the set of rationals with induced euclidian topology, let $\sim_1$ be the relation on $\mathbb Q$ which identifies all the integers, and let $\sim_2$ be the identity relation on ...
5
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1answer
477 views

Give examples of clopen (open and closed) sets

I was wondering whether somebody could help with the following problem. I think I have an answer, but I'm still a bit unsure so any help would be greatly appreciated. (a) Give an example of a ...
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2answers
36 views

examples of the lower semicontinuous functions

The following is the definition of lower semicontinuity: Let $T$ be a topological space and $f:T\to\Bbb{R}\cup\{+\infty\}$ a function. $f$ is said to be lower semicontinuous if the set $\{x\in ...
3
votes
2answers
174 views

Group Theory : A property related to the intersection of all subgroups

I was solving a problem from the book by Herstein (problem 2.5.2) and came across the following solution (the question is given there itself): ...
3
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1answer
44 views

A non trivial counterexample in three-space property

We know that completeness is a three-space property: Let $M$ be a closed subspace of a normed space $X$. Then, $X$ is complete if and only if $M$ and $X/M$ are complete. I am looking for a non ...
0
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1answer
35 views

Give an example of a function whose behavior becomes more erratic toward some limit.

Is there a (preferably simple) function that becomes more and more erratic as we take the variable to some limit? So suppose the limit is zero. Each time the function is evaluated with a smaller ...
2
votes
1answer
99 views

An unconventional algebraic function

We all know that there are algebraic numbers that can't be expressed by radical. For example the real root of the equation $x^5-x+1=0$ (which is near $-1.16$) is algebraic but can't be expressed by ...
5
votes
1answer
131 views

Example of non-homeomorphic spaces $X$ and $Y$ such that $X^2$ and $Y^2$ are homeomorphic

Can anyone suggest an example of non-homeomorphic spaces $X$ and $Y$ such that $X \times X$ and $Y \times Y$ are homeomorphic?
2
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3answers
213 views

Uncountable Ring with Finite Characteristic

Any good examples of these? Countable is easy, and uncountable is easy if I don't care about the proof being constructive but I really want something I can get a solid grip on. So nothing requiring ...
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2answers
47 views

Sequence of continuous functions converging pointwise to a non-continuous function [closed]

Is it possible that sequence of continuous functions is pointwise convergent to a non-continuous function?
3
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4answers
83 views

Is it possible to construct a metric in $\mathbb{R}^n $ s.t. it does not induce CONVEX balls?

I'm studying point set topology and looking for a counterexample of "Balls are convex". We say set $K \subset \mathbb{R}^n $ is convex if $\forall x, y\in K$ implies $\lambda x + \left(1-\lambda ...
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0answers
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Schröder-Bernstein for abelian groups with direct summands

What is a simple example of two abelian groups $A,B$ which are isomorphic to direct summands of each other (that is, $A \cong B + C$ and $B \cong A + D$ for some abelian groups $C,D$), but which are ...
0
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1answer
31 views

What is an example of a function continuous at point but discontinuous on any $\epsilon$-balls?

Let $E$ be open in $\mathbb{K}^n$ and $V$ be a normed space over $\mathbb{K}$. Let $f:E\rightarrow V$ be a function which is continuous at a point $p\in E$. Then, is it possible that for any ...
0
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1answer
23 views

Is there an example of a topological space that is of the Second Baire Category but is not a Baire space?

By being of the second category I mean that it is not the countable union of nowhere dense sets and by Baire space I mean a space such that a countable intersection of open dense sets is dense in X. ...
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1answer
43 views

Looking for example of topological space , not co-countable topology , such that closure does not imply sequential closure

I am looking for example of a topological space $X$ , whose topology is not the co-countable topology , such that there is some $A \subseteq X$ such that there exist some $a \in \bar A$ but there is ...
2
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2answers
137 views

Examples of monotone functions where “number” of points of discontinuity is infinite

We know that if $f:D(\subseteq \mathbb{R})\to\mathbb{R}$ be a monotone function and if $A$ be the set of points of discontinuity of $F$ then $\left\lvert A \right\rvert$ is countable. Where ...
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2answers
26 views

Relation between Basis of a Vector Space and a Subspace

$V$ is a vector space, and $H\subset V$ is a subspace of $V$. If $\beta$ is a basis for V, can we guarantee that $\beta \cap H$ is a basis for H?
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1answer
98 views

Give the example of compact set with infinite countable derived set [closed]

Can anyone give me an example of compact set of which the derived set is infinitely countable set?? thks in advance, I have no idea about this .
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0answers
21 views

Image is not a manifold when considered as a subset: how is this possible?

Wikipedia offers two definitions of a submanifold: One is that it is the image of an immersion. But I can't make sense of the remark that " in general this image will not be a submanifold as a ...
5
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3answers
85 views

Categories with some but not all exponentials

The introductory examples typically given of exponential objects in categories in fact involve categories which have all exponentials. Are there not-too-esoteric examples of categories of ...
2
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1answer
82 views

Is a Noetherian normal local domain (universally) catenary?

Let $R$ be a ring. Then $R$ is $\textit{catenary}$ if for a pair of prime ideal $p \subseteq q$, all maximal chains of prime ideals $p = p_0 \subseteq p_1 \subseteq \dots \subseteq p_n = q$ have the ...