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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3
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4answers
147 views

Number of elements in a group and its subgroups (GS 2013)

Every countable group has only countably many distinct subgroups. The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have ...
1
vote
2answers
60 views

Probability of result

Morning, A inbound contact centre says they will answer calls within 3 mins 70% of the time, if I ring the call centre 5 times over a week what is the probability I get my calls answered in under 3 ...
2
votes
1answer
299 views

Existence and uniqueness of limit of inverse function

Let $f:(a,b) \rightarrow \mathbb{R}$ be a one to one function. If $x_0$ is a point of the open interval $(a,b)$ such that $\lim_{x \rightarrow x_0} f(x) = l$, is it necessary that $\lim_{x \rightarrow ...
2
votes
2answers
114 views

Continuous bijection whose inverse is not continuous at uncountably many points

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible ...
1
vote
1answer
42 views

Let $f\geq 1$, Is the function $p\rightarrow \int |f|^p d\mu$ continuous

Let $f:X\rightarrow [0,\infty[$ be a measurable function that is greater than or equal to $1$ for every $x\in X$ and $\mu$ be a positive measure on $X$. Consider the function $g:]0,\infty[\rightarrow ...
3
votes
1answer
206 views

Applications of the Pontryagin product for abelian groups

For an abelian group $G$, one can give an explicit description of the homology ring $H_*(G, k)$ for e.g. $k=\mathbb{Q}, \mathbb{Z}_p$ or in general PIDs $k$ in which every natural number is ...
16
votes
1answer
385 views

Examples of universal constructions in probability theory

I am looking for more examples of universal constructions in probability theory. Like the construction a of Gaussian space from a real Hilbert space, or a Poisson jump process from a measurable space ...
1
vote
1answer
67 views

Closed sets, boundary, topology.

Let A be a closed subset of the real numbers. It is always possible to find a subset B of the real numbers such that A is equal to the boundary of B? Prove if true, find a counterexample if not. I ...
2
votes
1answer
59 views

Submonoids of $\mathbb{N}^k$

Do you know if all submonoids of $\mathbb{N}^k$ are finitely generated? If not, can you give me a counter-example? EDIT : I mean $\mathbb{N}^k$ as a submonoid of $(\mathbb{Z}^k,+)$. I already know ...
6
votes
1answer
220 views

Does “locally connected and path-connected” imply locally path-connected?

Some friends and I discovered this question when we were studying for an exam and were trying to find examples for all combinations of topological properties we had seen in the course so far. One we ...
0
votes
2answers
34 views

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$ Where $\Omega = \{(x,y,z) : z \leq 1\}$ and $\rm{Fr}(\Omega)$ means the $\Omega$'s boundary.
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votes
3answers
4k views

Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
0
votes
2answers
149 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
1
vote
2answers
53 views

Example of a continuous bijective function on and to the closure of the complex numbers, with an inverse that is not continuous?

Note that by the closure of the complex number, I mean the union of the complex numbers and infinity. I have been stumbling over this questions for a wile now, and I understand many examples of this ...
13
votes
1answer
204 views

What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
4
votes
1answer
93 views

Ultraproducts by countably complete ultrafilter

I've recently learned about ultraproducts, but the source I learned from almost immediately after the definitions restricted to talking about countably incomplete ultrafilters. I know that the ...
1
vote
1answer
18 views

Show that there exitst $f \in L^{1}([0,1])$ such that $\int_{0}^{1}f(x)g(x)dx \nrightarrow 0$

Define $$ g_{n} = n\mathbb{I}_{[0,\frac{1}{n^3}]}(x)\;\; $$ where $\mathbb{I}$ is index function. (if $x \in E, \mathbb{I}_{E}(x) = 1$, otherwise 0) show that there exists $f \in L^{1}([0,1])$ such ...
1
vote
1answer
269 views

Prove by minimum counterexample that $2^n>10n$ for $n>5$

Prove by minimum counterexample that for all integers $n>5$ the statement $2^n>10n$ is true. Attempt: Let $S$ be a set of counterexamples, $S=\{n \in \mathbb{Z_+}: 2^n \le 10n, \space ...
2
votes
1answer
129 views

A question on spaces with a point countable base?

Is there a normal Hausdorff space with a point countable base and a dense Lindelöf subspace which is not second countable? Thanks for your help.
1
vote
1answer
135 views

Repetitive tiling implies finite local complexity

My question probably needs to include the definitions of the terms in the title so I will first ask the question and then introduce the necessary definitions. The following Theorem is stated without ...
3
votes
2answers
80 views

For any element $e$ of an open set $V$ of a covering space, does there exist a sheet $S$ such that $e\in S\subseteq V$

Let $p:E\rightarrow X$ be a covering map. Let $V$ be any open subset of $E$ and $e$ be any element of $V$. I feel that the following statement must be true: There exists an evenly covered open subset ...
2
votes
1answer
216 views

Flatness, Hilbert polynomial and reduced schemes.

Let $f:X \to S$ be a projective morphism of schemes and $F$ a coherent $O_X$-module. We have that if $F$ is $S$-flat then the Hilbert polynomial $P(F_s)$ is locally costant as a function of $s \in S$. ...
2
votes
1answer
39 views

Does there exist a reversible monoid that fails to be Dedekind-finite?

Call a ring with unity reversible iff $xy = 0$ implies $yx = 0$. Dedekind-finite iff $xy = 1$ implies $yx = 1$. It is proved here that every reversible ring is Dedekind-finite. Now clearly, the ...
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0answers
92 views
3
votes
1answer
716 views

Separable metric space has a countable base

A collection $\{V_{\alpha}\}$ of open subsets of $X$ is said to be a $\textit{base}$ for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have ...
0
votes
1answer
70 views

Counterexample to: if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ with $\mu(X)=\infty$

We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality). Is this still true if $\mu(X)=\infty$? Counterexample? ...
-1
votes
3answers
191 views

Linear algebra. Find a counter-example

this is the statement: if $\vec v_{1}, \vec v_{2} , \vec v_{3}, \vec v_{4}$ is a basis for the vector space $\Bbb R^{4} $, and W is a subspace of $\Bbb R^{4}$, then some subset of the $\vec v$ 's is a ...
3
votes
1answer
59 views

Irreducibility of Lie algebra representations

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and $\pi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a homomorphism of real Lie algebras where $V$ is a finite dimensional real vector space. But ...
2
votes
1answer
194 views

$f$ not measurable, but $\lvert f\rvert$ measurable

Do you know an example of a function $f\colon\mathbb{R}\to\mathbb{R}$ which is not $\mathcal{B}$-measurable but $\lvert f\rvert$ is $\mathcal{B}$-measurable?
7
votes
1answer
560 views

Dini's continuity vs Holder continuity

(listed items are just the definitions, you can skip to "Clearly" if you are familiar with them) Let $E \subset \mathbb{R}^N$ and let $f \colon E \to \mathbb{R}.$ The modulus of continuity of $f$ is ...
6
votes
2answers
488 views

Operator with invertible adjoint

Let $X, Y$ be normed spaces and $T : X \to Y$ bounded and linear, such that its adjoint $T^* : X^* \to Y^*$ is boundedly invertible. If $X$ and $Y$ are Banach spaces, then $T$ is also boundedly ...
9
votes
2answers
4k views

What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
3
votes
1answer
55 views

Let $G_1,G_2$ be groups with 2 subgroups respectively $H_1,H_2$ satisfying certain conditions, must $|G_1:H_1|=|G_2:H_2|$

Let $G_1,G_2$ be groups with two subgroups respectively $H_1,H_2$ such that there is a bijection $f:G_1\rightarrow G_2$ and $f|H_1$ is a bijection between $H_1,H_2$. Must $|G_1:H_1|=|G_2:H_2|$ ? ...
4
votes
1answer
178 views

Counterexample to second derivative test when f''(x) is not continuously differentiable

When looking over true/false questions on previous midterms, one of my conscientious students said: "If f is defined on an open interval containing c, f'(c)=0, and f''(c)>0, then c is a local min of ...
6
votes
1answer
255 views

Existence of minimal non-zero prime ideals: Counter examples?

Let $R$ be an integral commutative ring with unit. If $R$ is noetherian, then every ideal has finite height, in particular, there exist minimal non-zero prime ideals if (and only if) $R$ is not a ...
1
vote
1answer
281 views

Example to $\lim f(x)g(x)$ may not exist

Let $A\subset\mathbb{R}$, $c$ a cluster point of $A$ and $f,g:A\rightarrow \mathbb{R}$. Suppose that $f$ is bounded on some neighbourhood of $c$ show by example that if $\lim_{x\rightarrow c}g(x)$ ...
0
votes
3answers
191 views

A finite dimensional vector space that is not naturally isomorphic to its dual.

I need an example of a finite dimensional vector space $V$ that is not naturally isomorphic to $V^\ast$. I know that, at least in finite dimensional case, there is a one-to-one correspondence between ...
2
votes
0answers
98 views

Is there a “natural” norm in the space of all sequences?

Let $X$ be $\mathbb{R}^\mathbb{N}$, the space of all real-valued sequences. Is there a natural norm in this space?
1
vote
0answers
34 views

Another question on spaces with calibre-$\aleph_1$

Let $X$ be a strongly monotonically monolithic space with calibre-$\aleph_1$. Must $X$ be Lindelof? I know $e(X)=l(X)$ for a strongly monotonically monolithic space. So to prove that $X$ is ...
4
votes
1answer
56 views

A question on spaces with calibre-$\aleph_1$

Suppose that $X$ is the $T_1$ space with $k$-in-countable base and $\aleph_1$ is a caliber of $X$. Must $X$ be second countable? Thanks for any help. A topological space has calibre $\aleph_1$ if ...
1
vote
1answer
47 views

Counterexample to $\operatorname{Supp} \mathcal G \subset Z \hookrightarrow X$ but $\mathcal G\cong i_*i^{-1}\mathcal G$ when $Z$ is not closed

Let $\mathcal G$ be a sheaf on a topological space and $X$ (say a sheaf of sets) and suppose its support is contained in a subset $Z$ of $X$, i.e. $\operatorname{Supp} \mathcal G \subset Z ...
2
votes
1answer
66 views

Another question on second countable spaces

Let $X$ have countable chain condition and point countable base. Is $X$ second countable? I thing it don't need to be. However I have no examples at hand. Thanks for your help.
3
votes
1answer
69 views

Seeking counterexample or proof for equivalence of two separation properties

Two points $x,y$ of a topological space are said to be distinguishable if at least one has a neighborhood not containing the other, and separated if each has a neighborhood not containing the other. ...
0
votes
1answer
91 views

A counter example about fraction of 2 sets

In a software company there are 2 departments, called A and B. In department A, the fraction of senior programmers (of out the junior programmers in this department) finishing the jobs in time is ...
0
votes
1answer
324 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
4
votes
1answer
84 views

Examples of spaces in which the closure of any path connected set is path connected.

Are there (non-trivial) examples of topological spaces in which the closure of any path connected set is path connected? If so, are there any far reaching topological consequences of this property? ...
5
votes
1answer
2k views

Construct a monotone function which has countably many discontinuities

I read in a textbook, which had seemed to have other dubious errors, that one may construct a monotone function with discontinuities at every point in a countable set $C \subset [a,b]$ by enumerating ...
0
votes
1answer
216 views

Explicitly showing cokernel of exponential sequence is not a sheaf

In the classical example of short exact sequence of presheaves $$0\rightarrow \mathbb Z \rightarrow \mathcal O_X \rightarrow \mathcal F \rightarrow 1,$$ it is well known that $\mathcal F$ is not a ...
0
votes
1answer
83 views

Counterexample to in $\mathcal{Mod}_A$ colimits of filtered index categories are exact

This is not a true general fact for any abelian category, as Vakil points out in 1.6.12. He gives the following counterexample, which puzzled for it is in the category of abelian groups, and every ...
3
votes
1answer
180 views

The product of limit point compact Hausdorff spaces is not limit point compact

Let $X, Y$ be limit point compact Hausdorff spaces (to be clear, a space is said to be limit point compact if every infinite subset of it has a limit point). Is it true that $X \times Y$ is limit ...