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)

1
vote
1answer
153 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
2
votes
1answer
117 views

A counterexample to $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$

Let $e,e^{\prime}$ be two idempotents in a $k$- algebra $A$ ($k$ is a field) . Then my guess $Ae\simeq Ae^{\prime}$ (as a left $A$- module) does not imply $A(1-e)\simeq A(1-e^{\prime})$ in general, ...
0
votes
2answers
44 views

A function with a finite limit at (both) 0 and infinity

I ask myself if there exists a $f(x)$ function that limit goes to a finite number for both when x goes to 0 and to infinity. Is it possible in some way ?
0
votes
2answers
321 views

Give a counterexample to show that $(AB)^{-1} \neq A^{-1}B^{-1}$

Give a counterexample to show that $(AB)^{-1}$ doesn't equal $A^{-1}B^{-1}$ I'm not sure how to approach this, so I just used the idea that the matrix multiplication is not commutative. so it goes: ...
1
vote
3answers
73 views

$f_n \to 0$ $ a.e.$ and $\lim \int f_n d\mu =0$ but $\sup_n f_n$ is not in $L^1$

Give an example of a finite measure space $(X,M,\mu)$ and a sequence of functions $f_n:X \to[0, \infty)$ such that $f_n \to 0$ $a.e.$ and $\lim \int f_n d\mu=0$ but $\sup_n f_n$ is not in $L^1$ I ...
0
votes
1answer
28 views

Show f need not be continuous.

If the functions $f_k$ are lower semicontinuous, $f_k \rightarrow f$ pointwise, and $f_{k+1} \geq f_k(x)$ then $f$ is lower semicontinuous. Show that $f$ need not be continuous even if the $f_k$ are ...
2
votes
0answers
58 views

Find counter-example about product measure

This problem is in Real analysis for graduate students, Richard F. Bass. Problem is the following Let $(X,\mathcal A)$, $(Y,\mathcal B)$ be two measurable spaces and let $f \ge 0 $ be measurable ...
0
votes
1answer
39 views

Is $N_G(S)=N_G(\langle S\rangle)$?

Related question: What would be a counterexample of $N_G(T)\not\subset N_G(S)$? Let $G$ be a group. Let $S$ be a subset of $G$. Then, is $N_G(S)=N_G(\langle S\rangle)$? I have proved that ...
7
votes
1answer
75 views

Is a topological space determined by its components and their quotient?

Given connected topological spaces $X_i$ and a totally disconnected space $Y$, is there a unique topological space $X$ with components homeomorphic to $X_i$ and $X/\sim$ homeomorphic to $Y$? ($\sim$ ...
1
vote
1answer
27 views

What would be a counterexample of $N_G(T)\not\subset N_G(S)$?

Let $G$ be a group and $S,T$ be subgroups of $G$ such that $S\subset T$. Is there an example such that $N_G(T)\not\subset N_G(S)$? Also, what is an example such that $N_G(S)\not\subset N_G(T)$?
6
votes
3answers
670 views

Some examples of virtually cyclic groups

The only virtually cyclic groups (ie. groups containing $\mathbb{Z}$ as subgroup of finite index) I really know are : the groups $F \times \mathbb{Z}$, where $F$ is a finite group, and the infinite ...
2
votes
1answer
114 views

Nowhere differentiability of Weierstrass function

It's again from Tao's book. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic interval ...
4
votes
2answers
70 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
2
votes
1answer
41 views

Does such a finite monoid exist?

Does there exist a finite monoid $M$ such that for some $x \in M,$ the following hold? $x$ cancels on both the left and the right: $$\frac{ax=bx}{a=b}\qquad \frac{xa=xb}{a=b}$$ $x$ has no two-sided ...
3
votes
2answers
224 views

Example of a normal extension.

Can you give an example of a Normal extension which is not a splitting field of some polynomial.? I know that splitting field of a polynomial is always a normal extension but i am looking for the ...
4
votes
3answers
412 views

Prove or disprove that if $f$ is continuous function and $A$ is closed, then $\,f[A]$ is closed.

If $f:X\to Y$ is a continuous function and $A\subset X$ is a closed set, then $f[A]$ is also closed. I know that if $f[A]$ is closed implies $A$ is closed then $f$ is continuous, but I'm not sure ...
-2
votes
2answers
6k views
1
vote
2answers
113 views

Finding a function $\mathbb{N} \to \mathbb{N}$ that is surjective but not injective.

The mapping is supposed to be from $\mathbb{N}$ to $\mathbb{N}$. I'm still trying to understand if this is possible, I mean if it was from $\mathbb{R}$ to $\mathbb{N}$, I guess $x^2$ would work.
6
votes
1answer
211 views

The strong operator limit of a sequence of unitary operators

If $\mathcal H$ is a Hilbert space and $U_n \in B(\mathcal H)$ is a strong-operator convergent sequence of unitary operators, say $U_n\rightarrow U$, is it true that $U$ is unitary? More explicitly, ...
0
votes
1answer
56 views

Finding a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$.

Suppose $f,g:[a,b]\to \mathbb R$. Provide a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$. I've been attempting to find a counterexample by ...
7
votes
1answer
211 views
1
vote
1answer
67 views

Counterexample of “the product of open subsets is open in a topological ring”?

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...
2
votes
3answers
97 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) ...
2
votes
2answers
215 views

Introductory example(s) of a functor that is full but not faithful

What is your favourite example to offer real beginners of a functor which is full but not faithful?
2
votes
1answer
104 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
2
votes
1answer
147 views

Maximal monotone operator without convex domain?

I'm looking for an example of a (multi-valued) maximal monotone operator $A$ mapping a Banach space $X$ into its dual $X^*$ such that the domain $D(A)=\{x\in X: Ax\neq\emptyset\}$ is not convex. ...
0
votes
1answer
49 views

A function $f$ such that the limit of $f(x^2)$ exists but not $f(x)$.

I want to show a function $f$ such that $\displaystyle\lim_{x\to x_0}f(x^2)\in\mathbb{R}$ but $\displaystyle\lim_{x\to x_0}f(x)$ doesn't exist. I only need a suggest of such a function $f$. I can't ...
0
votes
0answers
26 views

Clarification on the definition of truth

So I am learning about the Godel's theorem. My instructor define truth and falsity as something arbitrary. he define $f$ to be true is $f(x)$ is $1$ and if $f(x) = 0$, it is false. I still dont have a ...
2
votes
0answers
44 views

Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
2
votes
2answers
81 views

Example of $\pi$-metrizable space

A tychonoff space $X$ is $\pi$-metrizable if and only if it has a $\sigma$-locally finite $\pi$-base. Please help me to find some example of $\pi$-metrizable space. Is it true that every ...
1
vote
1answer
608 views

Total Variation Measure: Definition?

The total variation according to Rudin is defined as: $$|\mu|(E):=\sup_{\bigcup_{k\in\mathbb{N}}E_k=E}\sum_k|\mu(E_k)|$$ where the supremum is taken over all countable partitions. Now I'm reading in ...
0
votes
3answers
76 views

Borel Sets which are not intervals

I am looking for an element of the Borel-sigma-algebra which is not an (open, closed, half-open,...) interval. Can someone provide any example or an algorithm to construct them?
0
votes
2answers
27 views

What would be an example that rank of a subgroup of a free group is greater than the rank of free group?

Let $G$ be a free group and $H$ be a subgroup of $G$. Then, $H$ is also a free group. Let $R_G,R_H$ be ranks of $G,H$ respectively. what would be an example that $R_H>R_G$?
4
votes
4answers
148 views

What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
2
votes
1answer
40 views

Constructing noncommutative nilpotent rings of given index

When I read about algebra I often see a certain disregard for examples or perhaps a disregard for a reader whose knowledge of examples is limited. When I'm interested in a property $p$ of an algebraic ...
1
vote
1answer
121 views

Atlases on the topological manifold $\mathbb R$

I have been trying to produce an example of two incompatible atlases on $\mathbb R$. But no success. Could someone help me please? All my example seem compatible. For example, $A_1 = \{((-\infty,1), ...
4
votes
1answer
60 views

One-sided and two-sided cancellability

Is there a semigroup $S$ with right- and left-cancellable elements but no elements cancellable from both sides? $s\in S$ is left-cancellable if for any $a,b\in S$ we have $$sa=sb\implies a=b$$ and ...
0
votes
1answer
63 views

Seeking a possible counterexample in probability.

I am trying to find a counterexample or prove the following: $\dfrac{Var\left(X_{n}\right)}{\left[EX_{n}\right]^{2}}\rightarrow0 , then \dfrac{X_{n}}{EX_{n}}\rightarrow1$ in probability. Assuming ...
1
vote
1answer
65 views

Product Hilbert Spaces that require completion?

My lecturer said that to make the Hilbert space $(H\oplus H',⟨\cdot,\cdot⟩ )$ we need to (1) make the cartesian product $H \oplus H' = H\times H'$, (2) give it an inner product, and - the confusing ...
4
votes
3answers
79 views

Does a symmetric matrix $A^2$ imply a symmetric $A$?

Does a symmetric matrix $A^2$ imply a symmetric $A$? Any help would be much appreciated.
0
votes
2answers
155 views

Suppose f : A ---> B and g : B ---> A are functions for which g o f = 1A…

If I were to suppose that $f : A \to B$ and $g : B \to A$ are functions for which $g \circ f = 1_A$, is $f$ always surjective and is $g$ always injective? How would I either prove this or counter it? ...
0
votes
1answer
31 views

Continuity and Subspace Topology

I think the first one is false. If we let $(-1/2, 1/2) \subset \Bbb R$ and $(0,1/4) \subset \Bbb R$, then for $f(x) = x$ defined on $[0,1) \subset M = \Bbb R$, we have $f^{-1}(-1/2, 1/2) = ...
0
votes
1answer
70 views

A net in a product space and its cluster point

Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces. And let $Z=X\times Y$ be the product space equipped with the natural product topology $\mathcal{T}_Z$ on $Z$. Then, let ...
1
vote
1answer
44 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
0
votes
1answer
45 views

Ring subset which absorbs but is not an additive subgroup

Are there any undegrad-level examples of ring subsets which possess absorbtion property (as in ideal definition) but are not ideals (i.e. are not additive subgroups)?
1
vote
1answer
56 views

Example of ring with two maximal ideals such that the char of the quotients is $0$, respectively $p$.

I am looking either for an example of a commutative ring with identity and two maximal ideals, such that the characteristic of one of the quotient rings is finite and the other characteristic is zero, ...
0
votes
1answer
33 views

Stuck trying to find unbounded $s_n$ with $\frac{1}{n}\sum_{k=1}^n s_k\rightarrow L$

I proved that if a sequence $(s_n)$ converges to a limit $s$ then so does its "average sequence," $(\sigma_n)$ with $\sigma_n=\frac{1}{n}\sum_{k=1}^n s_k$. I found a counterexample for the converse, ...
2
votes
1answer
139 views

Proving the derived set $E'$ is closed.

I was reading the proof in Rudin, but it uses the metric. Is this not true if $X$ is a general topological space and $E' \subset X$ (especially if it is not Hausdorff $T_1$)? I can't come up with a ...
3
votes
1answer
107 views

intersection of locally compact Hausdorff topologies.

Are there locally compact Hausdorff topologies $\mathcal T, \mathcal S$ on a set $X$, such that $\mathcal T\cap \mathcal S$ is a Hausdorff but not locally compact topology on $X$?
3
votes
1answer
116 views

How to construct a smooth curve whose range is dense in $\mathbb R^2$?

How to construct a smooth curve $f: \mathbb R \to \mathbb R^2$ whose range is dense in $\mathbb R^2$? Space-filling curves are well-known, but they cannot be smooth. The image of a smooth ...