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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55 views

Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
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1answer
63 views

Are primitive row stochastic matrices diagonalizable?

Let $A$ be an $n \times n$ matrix with real, non-negative entries. Assume $A$ is primitive, meaning there exists an integer $k$ such that $A^k>0$ (here the inequality means all entries in $A$ are ...
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2answers
42 views

Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$

Let $f$ be a mapping. Prove that the following three statements are equivalent. $f$ is continuous; $\forall A \subseteq X: f(\overline{A}) \subset \overline{f(A)}$; $\forall B \subseteq ...
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0answers
26 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
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1answer
37 views

A Real valued function which is discontinuous **only** on a given specific set.

Let $\mathbb{L}=\{x_n \ |\ n=1,2,3 \dots\}$ be a countable subset of $\mathbb{R}$. My aim is to construct a real valued function $f$ on $\mathbb{R}$ such that $f$ is discontinuous at every point ...
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2answers
65 views

Vector space without a scalar product

In linear algebra the terms vector space and scalar product always (at least for me) appear together. Can you give me an example of a vector space without a scalar product? Does the senescence Let V ...
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2answers
78 views

Can the derivative of an absolutely continuous real function have a simple discontinuity?

If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous). More specifically, ...
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1answer
54 views

Sets and Logic .. Disproving with counter example

Can anyone please give me an idea to disprove the following with a counterexample: $A , B , C$ are sets. If $A \times C = B \times C$ , then $A = B$. (Here $\times$ is a Cartesian product.) I tried ...
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2answers
62 views

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges?

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges? $x_n$ is said to be bounded if and only if it is bounded both above and below. I believe this to be false. My ...
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1answer
26 views

A function with only a partial derivative not Hölder-continuous

I'm looking for a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$ 1. $x\mapsto u(t,x)$ is $C^{2,\alpha}$; 2. $t\mapsto u(t,x)$ is $C^{1,\alpha}$; 3. $t\mapsto ...
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21 views

Examples of “finite character”

The property of being a linear ordering has finite character, i.e. a relation linearly orders a set if it linearly orders all of its finite subsets. This is a trivial corollary of the definition. ...
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80 views

A perfect Hausdorff space that is not metrizable.

Can anyone provide an example of a well-known topological space that has the following three properties: (1) It is perfect (contains no isolated points), (2) T2, and (3) not metrizable.
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1answer
24 views

Finding a $C^1$ function $F(x,y)$ that is not of class $C^2$

I am trying to find a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ of class $C^1$, but not of class $C^2$. Meaning that $\frac{\partial^2 F}{\partial x \partial y}$ won't be equal to ...
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1answer
61 views

Question on Taylor's theorem

Taylor's Theorem Let $\{c_n\}$ be a sequence of complex numbers. Let $R$ be the radius of convergence of $\sum c_n z^n$. Let $|b|<R$ and $f(x)=\sum c_n z^n$ on the disk $B(0,R)$. ...
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2answers
37 views

Can analytic function be represented as a taylor series?

Let $D$ be open in $\mathbb{K}$. Let $f:D\rightarrow \mathbb{K}$ be an analytic function. Then, $f$ is infinitely differentiable and $\forall x_0\in D$, there exists a neighborhood $N$ of $x_0$ such ...
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3answers
530 views

Does there exist a non-abelian group with roots?

I just learned the definition of a division group: An abelian group $G$ is called divisible if for every $x\in G$ and every $k\in\mathbb{Z}^+$, there exists $y\in G$ such that $y^k=x$ (or $ky=x$, ...
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2answers
36 views

A Tychonov space non-homogeneous wrt neiborhood bases cardinality

Is there a Tychonov space $(X,\mathcal T)$ with $a,b\in X$ such that $a$ has a countable neighborhood basis while $b$ does not have any countable neighborhood bases?
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2answers
58 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
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1answer
44 views

Sobolev embedding counterexample

I trying to find a counterexample to show that $$W^{1,p}(\mathbb{R ^n}) \nsubseteq C^{0,\alpha}(\mathbb{R^n}) $$ for $p>n$ and $\alpha > 1 -\frac{n}{p}$. No clue yet, thanks for your help.
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2answers
53 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( ...
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1answer
56 views

Are homeomorphic, Hausdorff topologies on a set equal?

Let $\mathcal T$ and $\mathcal S$ be two topologies on a set $X$ and $(X,\mathcal T)$ and $(X,\mathcal S)$ be homeomoric and compact and Hausdorff. Is $\mathcal S$ equal to $\mathcal T$? I know the ...
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1answer
91 views

Two collections of subsets of a group and their equality [closed]

Let $G$ be a group and $H\le G$ and let $$\mathcal T=\{B\subseteq G \mid (\forall b\in B)(bH\subseteq B)\}$$ and $$\mathcal S=\{B\subseteq G \mid (\forall b\in B)(Hb\subseteq B)\}$$ Is $\mathcal T$ ...
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1answer
54 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, ...
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1answer
204 views

Tensor product of injective ring homomorphisms

What is an example of two injective homomorphisms $R \to A$, $R \to B$ of commutative rings such that $R \to A \otimes_R B$ is not injective? Of course neither $R \to A$ nor $R \to B$ can be flat in ...
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1answer
26 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
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1answer
109 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
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0answers
55 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
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3answers
132 views

What are some examples of coolrings that cannot be expressed in the form $R[X]$?

Let $K$ denote a field. Then the polynomial ring $K[x]$ has the property that the sum of two units is either a unit, or zero. I'll bet there's heaps of other examples, though. So let a coolring be a ...
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1answer
88 views

Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$

I am unsatisfied with the answers here. (Half of which used algebraic methods despite being advised not to!) Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$ ...
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2answers
44 views

Is the function $x \mapsto \max\{f(x),g(x)\}$ always an automorphism if $f$ and $g$ are?

Consider a totally ordered set $X$ and a pair of automorphisms (i.e., order-preserving bijections) $f,g$ of $X.$ Then the function $$h:x \in X \mapsto \max\{f(x),g(x)\} \in X$$ is an order-preserving ...
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1answer
302 views

How to convert a problem to a stars and bars problem?

Continued question from here. With certain questions I have $x_i$ being constrained by various different inequalities, I want to know how to remove these from the problem, to bring me back to a ...
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2answers
49 views

Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
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3answers
960 views

How to use stars and bars(combinatorics)

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$ Where $x_i\in\mathbb{N}$ Is this the correct time to apply the method?
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0answers
51 views

Examples of inner forms

Let $G$ and $G′$ be two linear algebraic groups over a field $F$. From what I understand, $G$ is called an inner form of $G′$ if $G$ and $G′$ are isomorphic over a the (or an?) algebraic closure of ...
7
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1answer
96 views

A locally metrizable, Lindelöf Hausdorff space that is not metrizable

I am looking for an example of locally metrizable, Lindelöf Hausdorff space that is not metrizable. I've proved that if such space is regular, then it is metrizable. The proof relies on the ability ...
2
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1answer
88 views

Examples of reduced associative algebras

An associative $K$-algebra A is called reduced if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. ...
3
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1answer
81 views

Examples of division algebras

Together with the Grunwald–Wang theorem, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is cyclic, i.e. can be obtained by an explicit ...
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2answers
42 views

Connectedness and non-local-connectedness of a subspace of $\mathbb R^2$

Let $(X,\tau)$ be the subspace of $\mathbb R^2$ consisting of the points in the line segments joining $(0,1)$ to $(0,0)$ and to all the points $(1/n,0)$, $n=1,2,3,\ldots$. Show that $(X,\tau)$ ...
7
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1answer
131 views

Handwaving gone wrong

My motivation for this question is twofold: On one hand, I'm studying algebraic topology, where - at least in the book written by Hatcher - there is quite a lot of handwaving (e.g. maps are continous ...
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2answers
82 views

Metrizability of quotient spaces of metric spaces

Suppose $X$ a metric space and $\sim$ an equivalence relation on $X$. Is the space $X/\mathord{\sim}$ metrizable? I think that the answer is no, but I could not arrive at a counterexample.
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An example of $k$-independent distributions.

I'm trying to better understand the idea of $k$-independence in distributions. The idea is that a distribution $\mu$ over $\{0,1\}^n$ is $k$-independent if any restriction of $\mu$ to $k$ variables ...
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0answers
18 views

Inclusion of commutators on classical pseudodifferential operators

We denote by $Cl^\mu$ the class of classical pseudo-differential operators of order $\mu$. Consider the notation $$[Cl^{a},Cl^{b}]\hookrightarrow [Cl^{a'},Cl^{b'}]$$ which means that a commutator on ...
4
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1answer
48 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 ...
4
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2answers
90 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
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1answer
36 views

Converse to Exactness of localization of modules

It is a standard fact that if $R$ is a ring, and $$\tag{1} 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0 $$ a short exact sequence of $R$-modules, if $S$ is a multiplicative ...
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1answer
23 views

closedness of a sublattice under complements

Let $A$ be a bounded sublattice of the bounded lattice $(X,\le)$ with $$\max A=\max X, ~~\min A=\min X$$ Let $a,b\in X$ be complements and $a\in A$. Is $b\in A$?
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1answer
38 views

Nowhere dense set - coarser vs. finer topology

Let $X$ be a set and let $\tau_1\subseteq\tau_2$ be topologies on $X$. Suppose that $A\subseteq X$ is nowhere dense in $\left(X, \tau_2\right)$. I was wondering if it follows that $A$ is nowhere dense ...
4
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1answer
63 views

Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
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9answers
2k views

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
60 views

Need an example/counterexample of continuous and increasing function.

If $\mu$ is a finite measure on the measurable space $\big( X, \mathscr{F} \big)$, $f : X\to [ 0, +\infty)$ is measurable. Then $\textbf{does it exist a continuous function $g : [ 0, +\infty)\to [ ...