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.

4885 questions
3
votes
0 answers

Is there a "natural" norm in the space of all sequences?

Let $X$ be $\mathbb{R}^\mathbb{N}$, the space of all real-valued sequences. Is there a natural norm in this space?
3
votes
1 answer

AB does not imply KC

We say that a space $X$ is: 1)AB provided that $X$ is $T_1$ and for each pair $A,B$ of compact disjoint subsets of $X$ there is $U$ an open subset of $X$ such that either $A\subseteq U$ and $U\cap B=\emptyset$ or $B\subseteq U$ and $U\cap…
3
votes
1 answer

$\mathcal{C}$ a family of connected sets with connected union. For all $C \in \mathcal{C}$ there is $C' \in \mathcal{C}$ with $C\cup C'$ connected.

It is easy to show that if the union of a finite family $\mathcal{C}$ of (more than one) connected sets is connected, then for any $C \in \mathcal{C}$ there must always be some other $C' \in \mathcal{C}$ such that $C \cup C'$ is connected. In fact I…
Herng Yi
  • 2,966
  • 15
  • 30
3
votes
1 answer

Can we find an example of non-mesuarable set which their outer measure could be computed?

We know there is non-measuarable set and we know every set has outer measure, so can anyone give me an example of a non-measuarable and there outer measure could be computed ?
henry
  • 1,699
  • 10
  • 20
3
votes
4 answers

Boolean prime ideal theorem and the axiom of choice

The Boolean prime ideal theorem is strictly stronger than ZF, and strictly weaker than ZFC. I'm looking for nice examples (like the existence of non-measurable set) that request at least that theorem (ZF+BPI), but weaker than AC (ZFC). The context…
dtldarek
  • 36,363
  • 8
  • 53
  • 121
3
votes
1 answer

If $f(x)$ is positive and decreasing, can $xf(x)$ have more than one maxima?

Assume that $f(x)$ is positive and decreasing on $[0,1]$ with $f(0)=1$ and $f(1)=0$. We see that $xf(x)$ is 0 at 0 and 1 and is positive in between so it must have a maxima. Is it possible that $xf(x)$ has more than one maxima on $[0,1]$? This…
Martin Leslie
  • 1,316
  • 8
  • 23
3
votes
1 answer

Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff every convergent sequence of points of $X$ has a…
3
votes
3 answers

Counterexample for a complex analysis proof

I'm having troubles coming up with a counterexample for the following: If $|f(z)|$ is continuous at $z_0$, then the function $f(z)$ is continuous at $z_0$ for complex numbers. I know I need a $f(z)$ that is discontinuous at that $z_0$, where…
MathStudent
  • 149
  • 1
  • 3
  • 7
3
votes
1 answer

Give an example of a program in Simply Typed Lambda that produces Bottom.

I'm not sure how bottom applies to simply typed lambda calculus. not A is a common abbreviation for A -> ⊥ But I see no way to construct a function of that signature within the theory. Edit: A more precise way of phrasing my original question is "Is…
3
votes
3 answers

Can an object be universal with respect to several properties?

Does anyone know an example of an object $A$ in some category $\mathcal C$ such that $A$ satisfies at the same time more than one universal property? Of course, when I say that $A$ is universal with respect to a certain property, I am aware that $A$…
3
votes
1 answer

Examples of affine schemes

I have to introduce affine schemes as topological spaces in a small seminar. Could you suggest me three examples of affine schemes in order to put in evidence the correlation among traditional algebraic geometry and the geometry of schemes. For…
ArthurStuart
  • 4,604
  • 19
  • 46
3
votes
2 answers

A non-exponentially bounded analytic function?

A function $f:\mathbb R\to\mathbb R$ is said to be exponentially bounded if there is an $n$ such that for sufficiently large $x\in\mathbb R$, $\exp(\exp(\cdots \exp(x)))>f(x)$ (where the $\exp$ is repeated $n$ times). You know what analytic means. …
3
votes
2 answers

Groups with $\wedge$-irreducible trivial subgroup

Suppose $G$ is a group satisfying the following condition: $$H \cap K = \{1\} \implies H = \{1\} \;\text{ or }\; K=\{1\}$$ for any two subgroups $H$, $K$, i.e. the trivial subgroup is $\wedge$-irreducible in the lattice of subgroups $L$ of $G$. I…
Luca Bressan
  • 6,625
  • 2
  • 18
  • 44
3
votes
1 answer

A counter example of best approximation

Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm $||f||=\max\limits_{x\in[a,b]}|f(x)|$. Let $B$ be a…
3
votes
1 answer

Seeking counterexample or proof for equivalence of two separation properties

Two points $x,y$ of a topological space are said to be distinguishable if at least one has a neighborhood not containing the other, and separated if each has a neighborhood not containing the other. It is readily seen that in an $R_0$ space (one in…
1 2 3
99
100