# Questions tagged [examples-counterexamples]

To be used for questions whose central topic is a request for examples where a mathematical property holds, or counterexamples where it does not hold. This tag should be used in conjunction with another tag to clearly specify the subject.

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### Surjective étale morphism between normal schemes

If I have $X,Y$ noetherian schemes, locally of finite type over a field $k$, normal and connected, can I somehow conclude that a surjective étale morphism $X \to Y$ is finite? I have seen some counter ...
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### If $x_n\ge 0$ for each $n\in\Bbb N$ and $\sum_{n\in\Bbb N}x_n=0$ then $x_n=0$ for each $n\in\Bbb N$ necessarly. [duplicate]

Let be $(x_n)_{n\in\Bbb N}$ a not negative sequence of real number and we suppose that $$\sum_{n\in\Bbb N}x_n=0$$ So if the last identity holds then it must be $$x_n=0$$ for any $n\in\Bbb N$ ...
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### $\mathbb{R}_l^2$ is not normal. [duplicate]

Let $\mathbb R_l$ be the Sorgenfrey line, i.e. $\mathbb R$ with the lower limit topology. I want to prove that $\mathbb{R}_l\times \mathbb{R}_l$ is not normal. I do not want to use Urysohn's lemma ...
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### Is there example of equalizer that is not injective?

we know that in general case every arrow is not function also in the $Div \mathbb{A}b$ there is example of monic that is not injective. Is there example of equalizer that is not injective ?
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### Quotients of real projective spaces

I am trying to determine whether or not $M := \mathbb{R}P^4/\mathbb{R}P^2$ is a manifold. I believe it is clear that $M$ is second-countable. However, I'm not not sure about Hausdorffness and being ...
1 vote
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### Counterexample for Mecke equation in higher dimensions

I am currently reading the book Lectures on the Poisson Process by Gunter Last and Mathew Penrose. (The book can be found here.) I have a question about an exercise in the book's 4th chapter (Exercise ...
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### Examples of counterintuitive regulated functions (solution verification)

Questions I am looking for some counterexamples for properties that of functions that are commonly mistaken for being true in cases where they do not hold in general. The specific cases I am thinking ...
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### If $K$ is an invertible matrix so $K+K^{-1}$ is diagonalizable, is $K$ necessarily diagonalizable?

Let $K$ be an invertible $n\times n$ matrix so $K+K^{-1}$ is diagonalizable. Is $K$ necessarily diagonalizable? Can't come up with a counter example, but also can't prove it... Here's my attempt so ...
1 vote
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### Is the boundary of singleton set in $T_1$ space is empty?

The singleton set in $T_1$ topological space is closed so, I wonder how i use this information to prove that boundary of this set is empty.
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### Can you disprove this counterexample to the diagonal lemma?

I was looking at the Diagonal Lemma or Fix point theorem which states in every Theory $T$ every formula with one variable $B(n)$ has a fix point: $T \vdash G \leftrightarrow B(\# G)$. Where $\#F$ ...
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### A sort of converse of Banach-Steinhaus theorem.

$(X, \|•\|)$ and $(Y, \|•\|')$ be two normed space. \begin{align} {\scr{B}}{(X, Y) }&=\{T\in {\scr{L}}{(X,Y)}: T \text{ is bounded } \}\end{align} $\|T\|_{op}=\sup\{\|Tx\|':\|x\|\le 1 \}$ ...
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### What are some good, elementary and maybe also interesting proofs by induction?

I am hosting a one-time class/talk on the concept of infinity for some (talented) high-school students. I want to teach them about proof by induction and I want them to do some exercises (you learn ...
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### Let $1=n_0<n_1<\ldots$ be an increasing sequence of positive integers. True/False: $\sum_{i=1}^\infty\frac{n_{i+1}-n_i}{n_{i+1}}$ diverges to $\infty$ [duplicate]

Let $1=n_0<n_1<n_2<\ldots$ be an increasing sequence of positive integers. Is it true that $\displaystyle\sum_{i=0}^{\infty} \frac{n_{i+1}-n_i}{n_{i+1}}$ diverges to $+\infty?$ For example, ...
1 vote
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### Are the sets $\Bbb R^+\times\{0\}$ and $\mathcal G(f):=\{(x,y)\in\Bbb R^2:xy=1\wedge x\in\Bbb R^+\}$ homeomorphic?

So I would like to prove that the sets $\Bbb R^+\times\{0\}$ and $\mathcal G(f):=\{(x,y)\in\Bbb R^2:xy=1\wedge x\in\Bbb R^+\}$ are homemorphic and to do this I tried to use the following ...
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### Example of an algebra that is a Banach space but not a Banach algebra

I'm looking for an example of a space $\mathbb{A}$ such, $\mathbb{A}$ is an algebra; $\mathbb{A}$ is equipped with a norm that makes it a Banach space; $\mathbb{A}$ is not a Banach algebra, i.e., ...
1 vote
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### ≞ (measured by symbol) example usage

I found this new equal sign that is named ' measured by ' (≞), and tried to find an exaple usage of it to no avail. I am a little uncertain about how to use it, and have two hypotheses. I think it is ...
1 vote
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### If $E,F\subseteq X$ are homeomorphic set then is $E$ open/closed if $F$ is open/closed?

Let be $X$ a topological space and we suppose that $E$ and $F$ are homeomorphic though a map $f$ from $F$ to $E$. So I ask to me if $E$ is open/closed when $F$ is open/closed but unfortunatley I was ...
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### Is uniform convergence required for a continuous limit function?

Consider the sequence $(f_{n})_{n=1}^{\infty}$ of continuous functions on $I = [0, \infty)$ defined recursively by $f_{1}(x)=x, f_{n}(x)=x+\int_{0}^{x}f_{n-1}(t)\sin(x-t) dt, \forall n\geq 2$. This ...
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### Finding an arbitrary pair of sets $U,V$ that is a separation of $A \in X$ and will satisfy $U \cap V \cap (X - A) \neq \emptyset$

Say I have some topology $\mathcal{T}$ on $X = \lbrace a,b,c \rbrace$ and a disconnected subset $A \subset X$. Can it be true that any arbitrary pair of sets - say $U$ and $V$ - that is a separation ...
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### Uniform continuity and derivatives [closed]

Is there any example of a three times differentiable and uniformly continuous function $f: \Bbb R \rightarrow \Bbb R$ such that $f^{(3)}$ is bounded but $f^{(2)}$ isn't?
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### Please explain these counter-interpretations to these Natural Deduction arguments

I have attached a screenshot of the arguments, but here they are written out just in case. I am able to prove when sentences are equivalent, but could somebody please explain the counter-...
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### Examples of continua that are contractible but are not locally connected at any point

A continuum is a compact, connected, metrizable space. What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of ...
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### Does every subalgebra of a finite dimensional commutative $k$-algebra has a complementary subalgebra?

Does every subalgebra $B$ of a finite dimensional commutative $k$-algebra $A$ has a complementary subalgebra (i.e. there exists a subalgebra $C$ such that $A=B\otimes C$)? The analogous statement for ...
### If $X$ is subset of $\mathbb{R}$, '$X$ is closed set' is equivalent to '$X=Y'$ for some countable set $Y \subset \mathbb{R}$'? [duplicate]
If $X$ is subset of $\mathbb{R}$, '$X$ is closed set' is equivalent to '$X=Y'$ for some countable set $Y \subset \mathbb{R}$'? ($Y'$ means the limit point's set of $Y$) In many example, I can find ...