Tagged Questions

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
16 views

How do I prove that this map is a homeomorphism?

Let $X$ be a topological space. Let $\{X_i\}$ be a family of mutually disjoint open subsets of $X$ such that $\bigcup X_i = X$. Let $a_i$ be a point of $X_i$ for each $i$. Consider a quotient map ...
1
vote
1answer
32 views

Zero sets in Mrówka spaces

For a maximal almost disjoint family $\mathcal A$ of subsets of $\omega$ we choose a set $\{x_A:A\in\mathcal A\}$ of distinct points not in $\omega$ and define $\Psi (\mathcal A)=\omega\cup \{x_A:A\in ...
1
vote
1answer
25 views

What are some examples of these kinds of commutative semirings?

What are some examples of commutative semirings such that the following hold? Multiplication is idempotent i.e. we have $xx=x$ for all elements $x$. Addition is not idempotent i.e. there is at least ...
1
vote
1answer
15 views

Examples: Non-piecewise-constant + Non-Measurable

Reference This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral Problem What are examples of real-valued functions: Bounded + Countably-Infinitely-Valued + ...
3
votes
3answers
67 views

Example of Hausdorff and Second Countable Space that is Not Metrizable

Does there exist topological space that is Hausdorff and second countable but not metrizable?
5
votes
2answers
81 views

Example (?) of a Banach space containing an uncomplemented copy of itself

I was wondering the following: Background Question: Does there exist a Banach space $X$ which contains a copy $X_0 \subset X$ of itself that is not complemented? By "$X_0$ is a copy of $X$", I ...
1
vote
3answers
90 views

First Countable Spaces are Hausdorff or Not?

Does first countable imply Hausdorff? If not, what is an example of first countable space that is not Hausdorff?
0
votes
0answers
21 views

Analytic function with inconsistent asymptotic behaviour on rays

Consider an function $f$, defined continuously on the closed upper half plane, and analytic on the upper half plane. Going along any ray from the origin that go strictly up (ie. not along the real ...
0
votes
1answer
25 views

Is this question wrong? Sequence of polynomial approximates $\sin$

Prove that there cannot be a sequence of polynomial $p_n$ converging uniformly to $\cos$ or $\sin$ on $\Bbb R$. Doesn't the Taylor series completely contradict this question?
0
votes
1answer
31 views

What do the operator semi-groups have to do with PDE's?

Can anybody please help me to understand what does the semi-group do with partial differential equations? We started this subject very recently and we are now in the proof of Hille-Yosida Theorem, ...
1
vote
0answers
16 views

Find ideal defining $Gr_2(\mathbb{C}^5)$ in Pluker embedding

Let $Gr_k(\mathbb{C}^n)$ the Grassmannian variety of $k$-planes in the complex space $\mathbb{C}^n$. We can consider the Pluker embedding $$ \mathcal{P}: Gr_k(\mathbb{C}^n) \to \mathbb{P}(\Lambda^k ...
2
votes
1answer
71 views

Just How Strong is Associativity?

A friend of mine is using a lot of Non-associative Algebra for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like "brackets ...
0
votes
1answer
25 views

Is a ring $R$ factorial $\iff$ $R[X]$ factorial?

Let $R$ be a factorial ring. Then, the polynomial ring $R[X]$ is factorial. I was wondering if the other direction also works (i.e. $R[X]$ factorial $\implies$ $R$ factorial)? If not, please give ...
3
votes
2answers
66 views

Does existence of non-trivial solution of $S(x,y,z) = 0, \; S(y,z,x) = 0, \; S(z,x,y) = 0$ implies existence of trivial solution at $x=y=z$ axis?

My question is following. Suppose that you have an implicit surface given by equation $S(x,y,z) = 0$ (if it matters, now $S(x,y,z)$ is a polynomial function). I'm interested only in $\mathbb{R}^3_{+}$ ...
2
votes
1answer
66 views

What would be an example such that $\langle a\rangle\langle b\rangle \neq \langle ab\rangle$?

Let $R$ be an rng (no unity). Define $IJ$ as the ideal generated by $\{ab:a\in I, b\in J\}$ for every ideals $I,J$ of $R$. Let $I=\langle a\rangle , J=\langle b\rangle $ be principal ideals. What ...
0
votes
2answers
22 views

Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
2
votes
0answers
42 views

Generalized Riemann Integral: Nonexample?

Definition Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. (In fact, a Hausdorff TVS should suffice.) Consider functions $F:\Omega\to E$. Define the generalized Riemann ...
2
votes
1answer
36 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
0
votes
1answer
21 views

Finding an specific counterexample of the interchanging of the limit and integral.

Hi I found the following exercise where I'm stuck, I'd appreciate any help. Thanks in advances Let $0\le f_n\to f$ and $\int f_n\to c>0$ pointwise. Show that $\int \lim f_n d\mu\in [0,c]$ and ...
0
votes
1answer
55 views

Prove or disprove regarding sequences

Question: Find a sequence $\displaystyle \{a_n\}_{n=1}^{\infty}$ such that $a_n\rightarrow 0$ and $n\left|a_{n+1}-a_n\right|\rightarrow \infty$. If no such sequence exists, prove it. My try: At ...
0
votes
1answer
62 views

Not Quite Metrization

Let's say I have a space $X$ with a function $d\colon X \times X \to \mathbb R$ that has the following 2 properties: $d(x,y)\ge 0$ for all $x$, $y \in X$ and $d(x,x) = 0$ for all $x$, $y \in X$. ...
0
votes
2answers
50 views

Fun results from modular arithmetic

I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups. I'm looking ...
-1
votes
0answers
18 views

theorem applicable in real function and not in complex function apart from MVT

please quote a theorem which is applicable to real valued functions and not to complex valued functions apart from MVT and ROLLES Theorem along with example
5
votes
0answers
80 views

Connected space such that (almost) no subspace is connected [duplicate]

Is there a connected space $(X,\tau)$ such that $X$ has more than $2$ points and the only proper connected subsets of $X$ are the singletons?
0
votes
1answer
23 views

What is an example of ideals such that $IJ\neq JI$?

Let $R$ be an rng. Let $I,J$ be ideals of $R$. What is an example of $IJ\neq JI$?
5
votes
1answer
75 views

Counter-example for abelian category that is not concrete

I am trying to figure out a counter-example for abelian category that is not concrete category. Does the category of representations over a quiver work? Thanks,
5
votes
1answer
179 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
0
votes
3answers
39 views

Interior of a set?

I'm trying to think if their is any topology for which this is false: If G is an open set, then G = interior(G) Can anybody think of anything? I'm pretty sure it's straight forward.
6
votes
1answer
136 views

Topological spaces with unknown fundamental group

Are there any well known topological spaces for which the fundamental group is not known yet?
1
vote
1answer
26 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 ...
3
votes
3answers
52 views

An example of a Banach space whose evaluation map is not surjective?

I have been giving the following corollary while studying functional analysis Let $X$ be a normed vector space. Then the evaluation map $$ev : X \to X'' , x \mapsto (f \mapsto fx) $$ is an ...
0
votes
2answers
36 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 ?
11
votes
5answers
1k views

Humorous integration example?

I was just reading though an introductory calculus book and it has the note: NOTE When integrating quotients, do not integrate the numerator and denominator separately. This is no more valid in ...
0
votes
2answers
74 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
51 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
20 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 ...
1
vote
2answers
36 views

Hilbert spaces of holomorphic functions

Could you please give me some examples of Hilbert spaces of holomorphic functions? Or even books or notes on Hilbert spaces of holomorphic functions? I need just a good number of examples and perhaps ...
2
votes
0answers
35 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
33 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 ...
1
vote
1answer
85 views

Prove using the smallest counterexample technique that: $\binom {2n}n\leqslant4^n.$

Actually what i know is that i must assume that this statement is false and then try to come up with non sense statement. Prove by the smallest counterexample technique the statement $$\binom ...
1
vote
1answer
24 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 $N_G(S)\not\subset N_G(T)$?
12
votes
0answers
200 views

Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
2
votes
1answer
48 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 ...
7
votes
3answers
226 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
2
votes
1answer
24 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 ...
2
votes
2answers
23 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 ...
1
vote
2answers
52 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.
1
vote
1answer
26 views

Question on abstract algebra about invertiblity?

Let $(G,*)$ be a group. Could anyone give me an example $a,b\in G$ such that $$a*b=e ~ and ~ b*a\neq e $$ Where e is the identity element. I would appreciate any help. Thanks in advance!
1
vote
4answers
66 views

What would be an example of a magma such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?

Let $(M,\cdot)$ be a non-associative magma. What would be an example of $M$ such that there exists $x\in M$ such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?
2
votes
0answers
35 views

Embedding vs embedding as a closed subset

A version of Whitney's Theorem state that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$ as a closed subset. Is there an example of $n-$manifold which can be embedded in ...