Questions tagged [counterexamples]
A counterexample is an example that disproves a mathematical conjecture or a purported theorem. For example, the Peterson graph is a counterexample to many seemingly plausible conjectures in Graph Theory.
279 questions
- Bountied 0
- Unanswered
- Frequent
- Score
- Trending
- Week
- Month
- Unanswered (my tags)
3
votes
1
answer
201
views
Example(s) of presheaves on a category C failing to discriminate between objects of a category D into which C maps
I’ve been trying to refine my intuition of the Yoneda Lemma, and in the process of doing so, I’ve thought a lot about the following situation. Suppose $F:C \to D$ is a functor between locally small ...
3
votes
1
answer
497
views
Example of connected, locally connected metric space that isn't path-connected?
I tried to construct a reasonable example for someone, meshing together various sorts of dust, but I either failed or wound up with sets whose separation properties are far too delicate for the ...
-5
votes
1
answer
266
views
Known examples of conjectures stated while suspected false, to invite counterexamples?
Sometimes, in computational or experimental mathematics, one faces statements that seem almost certainly false yet are not directly refutable by current methods or feasible computation.
In such cases, ...
7
votes
3
answers
1k
views
Peculiar exception in the number of distinct values taken by the sums of the 6th degree roots of unity
For a nonnegative integer $n,ドル let $N_n$ be the number of distinct values taken by the sums of $n$ 6th-degree roots of unity (with repetitions). First few counts are $N_0=1,ドル $N_1=6,ドル $N_2=19,ドル and ...
7
votes
0
answers
152
views
Can image closures of polynomial maps of affine spaces always be surjectively parametrized?
This is a crosspost from my previous math.SE post.
Consider an affine variety $X \subseteq \mathbb A^m$ (say, over $\mathbb C$) that is the image closure of a polynomial map
$$\phi \colon \mathbb A^n \...
6
votes
1
answer
291
views
What's the relationship between the Zariski and Scott topologies on the (reverse-ordered) spectrum?
I don't know anything about algebraic geometry. I was bored at work, reading nLab, and noticed that the Zariski topology and Scott topology are vaguely similar. Strictly $T_0,ドル and almost never ...
0
votes
2
answers
171
views
continuous, strictly increasing univariate real function with derivative 0 almost everywhere
Are there actually a strictly increasing continuous function from $\mathbb{R}$ to $\mathbb{R}$ with derivative of 0 almost everywhere ?
I tried to build one with three real sequences $a_n,ドル $b_n$ and $...
6
votes
0
answers
265
views
Around Chertanov's problem
Some time in the 70s Chertanov asked whether there is a compact ccc radial space which is not Fréchet (all spaces are assumed to be Hausdorff).
Fréchet means that every point in the closure of a set ...
5
votes
0
answers
421
views
Potential Errors in EGA Chapter 0ドル_{\operatorname{IV}}$ Regarding Formal Smoothness and Formal Etaleness
tl;dr: A collaborator of mine and I have found potential errors regarding formal smoothness and formal etaleness in EGA Chapter 0 and built a potential counter example. Is our counterexample correct ...
1
vote
0
answers
115
views
Epimorphisms with kernel pairs
I am a bit lost understanding some subtleties in various form of epimorphy.
The nLab reports that an effective epimorphism is one that coequalizes its kernel pair. A regular epimorphism is simply one ...
5
votes
1
answer
278
views
Example: Forgetful functor $\operatorname{CMon}(C)\to \operatorname{Mon}(C)$ is not fully faithful
In our class notes it says that for an $\infty$-category with finite products, the forgetful functor $\operatorname{CMon}(C)\to\operatorname{Mon}(C)$ is in general not fully faithful.
Let $C$ be an $\...
5
votes
1
answer
268
views
Stable infinite category counterpart of pathological behaviours around the AB3,AB4 and AB5 axioms of abelian categories
In his 2002 paper "A counterexample to a 1961 theorem in
homological algebra" (Invent. Math.) Amnon Neeman exhibited the infamous and scary example of a cocomplete abelian AB4 (colimits are ...
4
votes
2
answers
260
views
Convergence of the Cesàro mean of iterated continuous functions
Does anyone have a counter-example of the following statement :
Let $f : [0;1] \to [0;1]$ a continuous function w.r.t. the usual topology. Let $A_n(x) = \frac{1}{n} \sum_{k=0}^{n-1} f^k(x)$ for $n \ge ...
1
vote
2
answers
363
views
Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?
Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that
The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which ...
7
votes
1
answer
615
views
Is there a "Closure-of-Range Theorem" for Banach spaces?
The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*,ドル the four conditions:
$T(X)$ is $s$-closed; $T(X)$ is $...