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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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
230 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
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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 ...
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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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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
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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
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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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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 ...
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34answers
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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
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
23 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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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 ...
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1answer
28 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
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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 ...
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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 ...
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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 ...
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1answer
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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
79 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 ...
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1answer
46 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
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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 ...
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2answers
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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 ...
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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. ...
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1answer
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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. ...
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0answers
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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 $Y=f(X)$ ...
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4answers
119 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 ...
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4answers
634 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 ...
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2answers
183 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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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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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 ...
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1answer
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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
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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
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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 ...
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2answers
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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
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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
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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
58 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.
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An example of an ideal of order $12$

Provide an example of an ideal in $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$ that has order $12$, and indicate whether the ideal is a principal ideal (if it is, then identify the generator for the ...
2
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1answer
64 views

What would be an example of free module such that cardinality of linearly independent set is greater than the rank?

Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has the IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=\mathrm{rank}(M)$. Let ...
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0answers
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Request for counter examples in group theory

I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory). I am particularly interested in ...
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4answers
356 views

Example of an associative binary operation, without identities or inverses.

In essence, I am looking for an example of a semigroup or a semicategory (closure is not that important, but it is useful) that is NOT a monoid or category. Hopefully, there is a neat and ...
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10answers
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Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
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Hahn-Banach Theorem for finite dimensional vector spaces

I am studying the Hahn-Banach Theorem but I'm having a hard time following some of the examples presented in my notes, so I would like to first study some small examples which help me to get some ...
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0answers
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Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then ...
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2answers
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Is the axiom of choice really all that important?

According to this book: The Axiom of Choice is the most controversial axiom in the entire history of mathematics. Yet it remains a crucial assumption not only in set theory but equally in modern ...
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0answers
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Non homeomorphic spaces with same homology groups [duplicate]

Is it possible for two spaces X and Y to have the same homology groups with X not homeomorhpic to Y.
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2answers
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a discontinuous function the square of which is continuous

give an example of a discontinuous function the square of which is continuous. The domain is $[0,1]$. I tried to use the indicator function of rationals, but its square is not continuous. EDIT:I am ...
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1answer
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If all the numbers $(1^\alpha,\,2^\alpha,\,3^\alpha,\,\dotsc)$ are integer, then $\alpha$ is an integer.

A theorem of Siegel asserts that If $\beta>0$ and $2^\beta,\,3^\beta,\,5^\beta$ are integers, then $\beta$ is an integer. The following result is a beautiful consequence of this theorem ...