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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Premeasures: Inner Measure vs. Outer Measure

Problem Given a plain space $\Omega$ and a ring $\mathcal{R}$. (In fact, a semiring would do the job, too.) Consider a premeasure $\mu:\mathcal{R}\to[0,\infty)$. For simplicity, preextend this to: ...
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78 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 be sufficient.) Consider functions $F:\Omega\to E$. Define the generalized ...
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2answers
49 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
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Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$

I'm looking for examples of monoidal categories $\mathbf{C}$ such that one of the following two statements holds. For all objects $X$ and $Y$ of $\mathbf{C},$ $\mathrm{Aut}(X \otimes Y) \cong ...
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2answers
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Example of topological space where there is a point and a subset $A $: $x \in \overline A $, but no sequence in $A $ converging to $x $?

It is known that if a space $X $ is metricable then for any subset $A $ of $X $ and a point $x \in \overline A $ there is a sequence of points in $A $ converging to $x $. I wonder if there is an ...
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Examples: Non-Step & Non-Measurable

Reference This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral Problem What are examples of real-valued functions: Bounded & Non-Step & Non-Measurable ...
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Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$.

$F$ and $K$ are fields. Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$. I think I need to find a polynomial in $F(r^3)[x]$ that has $r$ as a root. I ...
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22 views

Dimension Field True/False.

I'm having trouble approaching how to determine truthfulness and falsehood of the following type of problems. $F$ and $K$ are fields. 1) Suppose that $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ ...
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Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$

Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$. I don't really even know where ...
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The set of differentiability of an extension from half-plane to the plane

This question is related to: Differentiability: Partially Defined Functions Consider a real-valued function $f:\mathbb{H}^2\to\mathbb{R}$. Are there some that admit no extension differentiable in ...
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28 views

Example of metric space completion

I'm looking for examples of noncomplete metric spaces and their completions. I know of some basic examples like completion of open intervals and rational numbers(both with the reals and p-adic ...
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38 views

Counterexample for “if every continuous function on $E$ is bounded, then $E$ is compact” [on hold]

Give me counter example for this false statement: "Every continuous function on the set $E$ is bounded this implies $E$ is compact".
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59 views

If two elements a,b have a gcd, do then a*t,b*t also have a gcd?

Let $M$ be a commutative cancellative monoid. For elements $a,b \in M$ a gcd of $a,b$ is an element $\mathrm{gcd}(a,b)$ with the (universal) property $\forall c \in M (c |\mathrm{gcd}(a,b) ...
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Example of a series parallel graph with toughness greater than $\frac{4}{7}$

Can anyone lead me to an example of a "more than $\frac{4}{7}$ tough series parallel graph"? Graph toughness is defined as $T = \min \left\{\frac{|a|}{\omega{(G\backslash A)}}\right\}$ over all ...
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34 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 ...
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Are there any countable Hausdorff connected spaces?

Do countable Hausdorff connected topological spaces exist?
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1answer
49 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 ...
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3answers
80 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?
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3answers
96 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?
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1answer
26 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 ...
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7answers
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Illustrative examples of a phenomenon in the logic of mathematical induction

It is said (and I myself have said) that in some cases the easiest way to prove a statement by mathematical induction is to prove a stronger statement by mathematical induction, because then one has a ...
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2answers
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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 ...
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23 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 ...
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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 $\mathbb{R}^3_{+}$ ...
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Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, i.e. ...
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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, ...
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1answer
26 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?
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0answers
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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 ...
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iid variables, do they need to have the same mean and variance?

If two random variables $x$ and $y$ are identical and independently distributed, do they need to have the same mean and variance? Can there exist a case where they are iid and still have different ...
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1answer
72 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 ...
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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 $x$, $y \in X$. ...
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31answers
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Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
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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 ...
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0answers
38 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 ...
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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 ...
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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
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1answer
37 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 ...
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1answer
56 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 ...
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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 ...
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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 ...
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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?
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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
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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 ...
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2answers
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Zero mean but not a martingale

I am looking for a simple stochastic process which has zero mean for all $t\geq0$ but it is not a martingale. I been looking in to local martingales but having trouble keeping the mean zero for all ...
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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 $IJ\neq JI$?
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1answer
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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,
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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.
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1answer
87 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 ...
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1answer
49 views

A polynomial algebra that is free as an $A$-module

I'm working through some problems when I stumbled across a question asking about conditions for when the polynomial algebra $k[x_1,\ldots,x_n]$ is also a free $A$-module, where $A$ is some ...
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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 ...