Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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 disagree with the actual situation that follows from the definitions. This tag should be used in conjunction with another tag to clearly specify the subject.

Do not use for any question containing an example or a counterexample. The question must specifically be about examples or counterexamples.

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An example of a non first countable Fréchet-Urysohn space?

As the head title says, I need a Fréchet-Urysohn space but not first countable, (on the way, a good Text book to follow). Thanks.
Jesús Pérez
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Is the classical Mrowka space $\Delta$-normal?

Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset V$, where $\Delta_X$ is the diagonal of $X$,…
Paul
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Is every Hilbert space an $L^2$ space?

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
Amr
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Counter example for absolutely continuous measure

I need a example for the following statement: "Given a pair of finite measures $(\mu,\nu)$ on a given measurable space $(\Omega, \mathbb{A})$ is said to have property $P$ if for every $\epsilon >0$ there exists a $\delta >0$ such that for all $A \in…
Bananafish
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Does $\phi\circ g$ convex imply $\phi\circ f\circ g$ convex?

Assume that $\phi$ is an increasing function and $g$ is a decreasing function with $\phi\circ g$ convex. If $f$ is an increasing convex function does this imply $\phi\circ f\circ g$ is a convex function? I came to this from an applied problem where…
Martin Leslie
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Is $f(A)\cap f(B)$ a subset of $f(A\cap B)$?

Let $X$ and $Y$ be sets and let $f\colon X\to Y$ be a function from $X$ to $Y$. If $A$ and $B$ are subsets of $X$, is it true that $f(A)\cap f(B)$ is a subset of $f(A\cap B)$? If so, prove your answer; otherwise, provide a counterexample. If we…
Nicole
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An isometrie $\varphi: S_1\to S_2$ which cannot be extended into distance-preserving map

I'm searching for an example isometrie $\varphi: S_1\to S_2$($S_i$ are regular surfaces) which cannot be extended into distance-preserving maps $F: \Bbb R^3 \to \Bbb R^3$. A reference or hint will help too. \color{gray}{No need the proof of…
Hoseyn Heydari
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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 $\lim_{x \rightarrow l} f^{-1}(x) = x_0$? My guess is no. I have…
Steve Pap
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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 that $f^{-1}$ fails to be continuous…
Prism
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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 that every submonoid of $\mathbb{N}$ is finitely…
Plop
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What function(s) can satisfy these conditions?

For what function (or functions) is the following true: 1) $f(x)$ is positive for $x>0$ 2) $\lim\limits_{x\to 0}{f(x)} = \infty$ 3) $\lim\limits_{x\to\infty}f(x) = 0$ 4) $\int_{0}^{\infty} {f(x)} dx = C$ 5) $f(x)$ is symmetric over $y=x$ 6) $f(x)$…
user474632
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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 $U$ of $X$ such that $p^{-1}[U]=\cup_{i\in I}…
Amr
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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 $s \in S$. (for reference…
ArthurStuart
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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 $|G_1:H_1|=|G_2:H_2|$ ? Note: If we only require that $G_1,G_2$ have the same…
Amr
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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 considered some counterexample only like $(\mathbb{Z}, +)$,…
Supriyo
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