# 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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### Are there examples where torsion subgroup of the first homology group is not $(\mathbb{Z}_2)^n$?

I've been trying to understand nontrivial torsion subgroups $G$ of the first homology group $H_1\cong\mathbb{Z}^s \times G$, where $s$ is the rank (first betti number) and G is a finite abelian group. ...
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### Attraction of events

I don't know if the next statement is true or false: Let $(\Omega,\mathcal{F},P)$ be a probability space and let $A,B$ and $C$ be events in $\mathcal{F}$ such that $P(A)>0$. If $P(B|A)>P(B)$ and ...
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### Example of Complex Pythagorean Triples

I am looking for example of a Pythagorean Triple with Gaussian Integers. I followed the links and looked at followings : Relation to Gaussian integers in https://en.m.wikipedia.org/wiki/...
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### If we remove the diagonal from $X\times X$, is it necessarily disconnected?

If $X$ is a compact, connected Hausdorff space, we know that the diagonal $\Delta_X=\{(x,x)\in X\times X\}$ is closed in $X\times X$ by Hausdorffness. But is $X\times X\setminus\Delta_X$ disconnected ...
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### Coming up with a counter example - calculus

I have to come up with a counter example for the following statement: Let $f$ be a function $f: [0,\infty)\longrightarrow R$, continuous and bounded. Prove that it receives either a minimum or a ...
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### $\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series .

Conjecture : $$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/1!+1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$ Where $a_n$ is an integer such that : $$0\leq a_n\leq n$$ Some arguments : ...
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### Building a function $f$ such that $\| f - f_n \|_{L^p(B(x,r) \cap \Omega)} \to 0$ as $n \to \infty$ and $f \in L^p_{\text{loc}}(\Omega)$.

Consider an arbitrary open set $\Omega \subset \mathbb R^n$ and an arbitrary element $1 \leqslant p < \infty$. Moreover, let $(f_n)_{n \in \mathbb N} \subset L^p(B(x,r) \cap \Omega)$ denote a ...
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### There is a metric space on $\mathbb{Q}$ such that this space be compact.

Prove or disprove: There is a metric space on $\mathbb{Q}$ such that this space be compact. I find some examples of non equivalent metric on rational numbers, euclidean metric, p-adic and discrete ...
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### Necessity of Hausdorff-ness in "continuous function determined by its values on a dense subset"

It's well-known that if a continuous function taking values in a Hausdorff space is uniquely determined by its specification on a dense subset of the domain. Now, I contemplate on the necessity of ...
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### Exercise 1.1.9 from West

Currently I am reading the first edition "Introduction to Graph Theory" by Douglas B. West. Exercise 1.1.9 states the following: 1.1.9. Suppose that $G$ is a simple graph having no vertex of ...
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### Can an infinite sum of non-computable numbers be computable, such that all finite sums of subsets of terms are non-computable?

Background In the following question, User1 asks whether an infinite sum of irrational numbers can be rational. Multiple answers1 indicate the answer to this question is 'yes'. For instance, Rasmus ...
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