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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1answer
33 views

Show that cauchy sequences are not topological terms, find a counterexample?

I need to show that the expression 'is a cauchy sequence' is not a topological expression. I could show this by finding a homeomorphic function $\phi$ such that a Cauchy sequence is mapped onto a ...
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0answers
39 views

Give an example of a locally compact Hausdorff space which is not compact.

Give an example of a locally compact Hausdorff space which is not compact. Would the following be valid? Could someone give an example using a different topology? $(\mathbb{R}, ...
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6answers
470 views

Is $A^2=I \implies A=\pm I$ necessarily true?

$A$ is $n\times n$ matrix. How to prove whether it is true or false $$A^2=I \implies A=\pm I$$ I was trying on $2\times 2$ case...multiplying general entries and then equating them to the identity ...
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4answers
2k views

Continuous function from (0, 1] onto (0, 1)?

Is there any continuous function defined on (0, 1] whose range is (0, 1)?
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1answer
304 views

A semisimple commutative Banach algebra with a non-semisimple quotient

I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient. Attempt from the comments: "I take $A$ to be the algebra of all continuously ...
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1answer
338 views

An equivalent definition of a group

Exercise 15 from Hungerford: Algebra. Let $G$ be a nonempty finite set with an associative binary operation such that for all $a,b,c\in G\,\,ab=ac \Rightarrow b=c$ and $ba=ca \Rightarrow b=c$. Then ...
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1answer
309 views

If product $X_1\times X_2$ and $Y_1\times Y_2$ are homeomorphic is $X_i\simeq Y_i$?

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...
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2answers
297 views

Topology on $\mathbb{R}$ strictly coarser (resp. finer) than the usual one which is still Hausdorff (resp. connected)

The following are simple observations. Suppose $\mathcal{T}_1,\mathcal{T}_2$ are two topologies on a set $X$ such that $\mathcal{T}_1$ is finer than $\mathcal{T}_2$. If $( X ,\mathcal{T}_2 ...
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2answers
56 views

In a metric space $(X, d)$, if closed sets $A$, $B$ contain sequences $a_n,b_n$ such that $d(a_n,b_n)\to 0$, must $A\cap B\neq \emptyset$?

I wrote a test yesterday in which one of the questions asked us to prove that if $A$ and $B$ are disjoint closed subsets of a metric space $(X, d)$, then there exist disjoint open subsets $U$ and $V$ ...
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5answers
618 views

Is the closure of a Hausdorff space, Hausdorff?

$(X,\mathcal T)$ is a topological space which has a dense Hausdorff subspace. Is $X$ Hausdorff?
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2answers
126 views

$\lim_{n\rightarrow\infty}n(\ln n)a_n=0$ implies $\sum a_n$ converges?

Is it true that If $$\lim_{n\rightarrow\infty}n(\ln n)a_n=0,$$ then the series $$\sum_{n=1}^\infty a_n$$ converges? If so, I want to know the proof. If not, I want to know the counter example.
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4answers
77 views
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2answers
45 views

Give an example of a topological space $X$, and a connected component which is not open in $X$

Give an example of a topological space $X$, and a connected component which is not open in $X$ I know of the following theorems: Each connected component is closed; If a topological space has a ...
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3answers
227 views

In a non-Hausdorff space, can a compact subset fail to be closed?

In a Hausdorff space $X$, every compact subset $Y$ is closed. So if I relax the condition on $X$ being Hausdorff, is it possible compact subset $Y$ of $X$ not being closed?
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4answers
14k views

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$. This question is just after the definition of differentiation and the theorem that if $f$ is finitely ...
4
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1answer
53 views

Is there a finite abelian group $G$ such that $\textrm{Aut}(G)$ is abelian but $G$ is not cyclic?

Is there an example in which $G$ is a finite abelian group and $\textrm{Aut}(G)$ is abelian but $G$ not cyclic?
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3answers
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If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
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1answer
1k views

Continuous Functions and Cauchy Sequences

We know that if a function $f: A \mapsto \mathbb{R}$, $A \subseteq \mathbb{R}$, is uniformly continuous on $A$ then, if $(x_n)$ is a Cauchy sequence in $A$, then $(f(x_n))$ is also a Cauchy sequence. ...
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0answers
25 views

Linear system without base points

Let $D$ be a divisor and $V$ a vectorial subspace of $L(D)$. The set of the divisors $S=\{(f)+D\mid f\in V\}$ is called a linear system. A point $P\in C$ is a base point if for every divisor $D\in S$, ...
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2answers
208 views

Space which is neither locally connected at any point nor totally disconnected

Let $X$ be a topological space; then we say that $X$ is locally connected at $x$ if $x$ admits a neighborhood basis of open, connected sets. In this sense, a space is locally connected iff it is ...
319
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34answers
19k views

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 ...
4
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1answer
70 views

Lindelöf property and $\omega$-covers

Let $X$ be a Lindelöf topological space. Does this imply that every $\omega$-cover has a countable subcover which is also an $\omega$-cover? if not, is there an example of a topological Lindelöf space ...
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1answer
22 views

Can a field extension still have “non-separability” above its maximal purely inseparable subextension?

Question 1 Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely ...
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0answers
47 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
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1answer
122 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this? I guess $A$ must be an $G_{\delta}$ set which is dense in $\Bbb ...
2
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1answer
26 views

Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
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0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
5
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3answers
34 views

Possible textbook redundancy concerning invertible mappings

In my textbook (Modern Algebra by John Durbin, 6th Ed), there is the following theorem: Let $S$ denote any nonempty set. (a) Composition is an associative operation on $M(S)$, with identity ...
8
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8answers
267 views

An example of a mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that $\eta(x)=n$ has infinitely many solutions for each $n\in\mathbb{N}$

Suppose I have the mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\eta(x)=n$ has infinitely many solutions. I saw this question which is basically the ...
3
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1answer
86 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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1answer
78 views

An example of a group G that satisfies $x^2 =e$ , for all $x \in G$

I thought for a long time about which group will have this property but I didn't get it. Is this a valid one so if G = {a,b} then ab = b and b is the identity for example then aa = b , ab = b and ...
1
vote
1answer
44 views

Locally Hamiltonian vector fields

Consider the following definitions (taken from [1]) Definition. Let $E$ be a Banach space and $B: E \times E \to \mathbb R$ a continuous bilinear mapping. Then $B$ induces a map $B^\natural: E \to ...
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2answers
24 views

Does one of these conditions on a sequence imply the other one?

Let ${(r_n)}_{n \geq 0}$ be a sequence of integers $\geq 2$. Set $q_n=\prod_{i=0}^{n-1} r_i$ (agreeing with $q_0=1$). I want to know whether one of these two conditions implies the other one (I think ...
1
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2answers
55 views

Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
5
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4answers
2k views

Examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical examples. ...
4
votes
1answer
71 views

Is it true that the intersection of a sequence $K_1 \supset K_2 \supset K_3 \dotsm$ of connected subsets of $\mathbb{R}^2$ is also connected?

I have got one counterexample for this : Consider the family {D} of closed discs centered at zero having radius $1+1/n$, i.e. disc $D_1$ has radius $1+1=2$, $D_2$ has radius $1+1/2=1.5$, and so on. ...
1
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0answers
32 views

Is imaged of a Polish Hausdorff space under an injecitve map always Hausdorff?

I have a question about Hausdorff topological space. Question: Let $X,Y$ be topological spaces. If $X$ is a Polish space (i.e. $X$ is a separable and completely metrizable space.) and $Y=f(X)$ ...
1
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4answers
116 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
3
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4answers
615 views

Continuous image of a locally connected space which is not locally connected

The question is pretty much in the title, I'm looking for an example of a locally connected space and continuous mapping such that the image is not locally connected. Thanks! EDIT: Corrected the ...
5
votes
2answers
182 views

The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $ 17 $ from Chapter $ 10 $ of the Fourth Edition of Michael Spivak’s Calculus. The statement is quite simple, but I have not had any success in finding an example. Here is the ...
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17answers
1k views

Accidents of small $n$

In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that ...
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2answers
261 views

Every subgroup of a normal subgroup is normal

Is every subgroup of a normal subgroup normal ? That is if $H$ is a normal subgroup of a group $G$ and $K$ is a subgroup of $H$, then $K$ is a normal subgroup of $G$. Is it true ? If not what is the ...
3
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1answer
140 views

Counterexamples to the Matrix norm AM-GM inequality?

I am new here and this my first question, I hope I am being as clear as possible and apologize in advance for any misunderstandings. I am researching the Arithmetic-Geometric Mean (AM-GM) inequality ...
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0answers
32 views

Simple examples of fractional ideals

Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$. ...
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1answer
47 views

How differentiable is the function $g(x) = \sum_n 2^{-n} f(x-r_n)$ where $f(x)=x^2 \sin\frac1{x}$?

This is an auxiliary enquiry (something like it may well be already discussed on MSE, but I haven't found it) resulting from a feeling of unease provoked by the question of this post. Taking the ...
1
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2answers
54 views

Cancellative Abelian Monoids

Is there an example of cancellative Abelian monoid $M$ in which we may find two elements $x$ and $y$ such that they have a least common multiple but not a greatest common divisor?
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1answer
40 views

Prove by counterexample that $\gamma$ and $\delta$ are not necessarily unique

Assume $\mathbb Q[\sqrt{d}]$ is a Euclidean Field and $\alpha$, $\beta$ are two quadratic integers in $\mathbb Q[\sqrt{d}]$, so that there exists integers $\gamma$ and $\delta$ in $\mathbb ...
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3answers
311 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
3
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1answer
41 views

Continuity of third derivative in extremum test

Consider the following standard real analysis textbook theorem: Let $I$ be a perfect interval, $f\colon I \to \mathbb{R}$ be $C^3$ (i.e. three times differentiable and $f'''$ continuous). If $x_0 ...
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1answer
57 views

Does an open map of the real line have to be one to one?

Is it true or not that open map $f:\mathbb{R} \rightarrow \mathbb{R}$ is a one one map? If it is one one explain how it is one one.