Comment by Gro-Tsen on Topological spaces in which countable intersections of...
@NotMike Indeed, I should have mentioned the P-space property. I added a remark to the question on this subject.
View ArticleComment by Gro-Tsen on Why are extremally disconnected spaces so hard to give...
@godelian This seems promising, and your example interests me. I can (sort of) form a mental picture of an $η_1$ totally ordered set, like the set of functions $ω_1 \to \{-1,0,+1\}$ that are eventually...
View ArticleAnswer by Gro-Tsen for Does the decomposability of $\mathbb{R}$ imply...
(To lighten notations, let me simply write $\mathbb{R}$, instead of $\mathbb{R}^d$, for the set of Dedekind reals in what follows, since it is the only one that will appear. For the sake of notational...
View ArticleWhat topos-theoretic construction lies behind the “symmetric model”...
Suppose we want to prove that (classical!) $\mathsf{ZF}$ does not prove, say, “for every infinite set $A \subseteq \{0,1\}^{\mathbb{N}}$ there exists an injection $\mathbb{N} \to A$” (I take this...
View ArticleEvery real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.Or so I'm told, but this leaves me...
View ArticleSimplicial complexes on $[n] := \{0,\ldots,n\}$ that are identical under any...
For $n\in\mathbb{N}$, let us denote by $\Omega(n)$ the set of all (possibly empty) “abstract” simplicial complexes on $[n] := \{0,\ldots,n\}$ (“on $[n]$” means “labeled by the elements of $[n]$”). To...
View ArticleAnswer by Gro-Tsen for Examples of common false beliefs in mathematics
I just learned incidentally in the comments of another question here that it is not true that every proper subgroup is contained in a maximal (proper) subgroup. A counterexample is easy to find: the...
View ArticleGraph chromatic numbers defined by interactive proof
Edit (2020-07-15): Since the discussion below is perhaps a bit long, let me condense my question to the followingShort form of the question: Let $G$ be a finite graph (undirected and without...
View ArticleWhat's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is...
View ArticlePreimage of a sublocale by a morphism of locales: description by nucleus?
For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs...
View ArticleAnswer by Gro-Tsen for How slow can an uncomputable function from...
Here's one possible interpretation of your question, I don't know if this is what you're after:Proposition. For every nondecreasing $b\colon\mathbb{N}\to\mathbb{N}$ such that $b(n) \to +\infty$ as $n...
View ArticleConstruction of the smallest nucleus above a prenucleus: what does this proof...
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is...
View ArticleAnswer by Gro-Tsen for Where does one learn about the weather?
I have myself only glanced through it, but I believe the book An Introduction to Dynamic Meteorology by James R. Holton (4th edition 2004) is a standard reference in the domain. It's more on the...
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Answer by Gro-Tsen for Is there a natural bijection from $\mathbb{N}$ to...
Since the Calkin–Wilf tree has been mentioned in other replies to this question, I think the Stern-Brocot tree needs to be mentioned as well, because (although it is very much related, basically up to...
View ArticleComputing the Heyting operation on the frame of nuclei
(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound...
View ArticleComment by Gro-Tsen on A topos for realizability under a variable oracle
@FrançoisG.Dorais Yes, I believe it's very different: the topos I'm trying to describe is supposed to “collect” the realizability topoi over $\mathcal{K}_1^x$, where $x$ ranges over Baire space, but...
View ArticleComment by Gro-Tsen on Removing disks with maximal radius in a disk: must the...
@JeanAbouSamra Isn't it a classic saying that the best part of emmenthal is the holes?
View ArticleEffective (algorithmic) computation of the moduli space of algebraic curves...
Much literature (see, e.g., references here) has been written about the existence and “construction” of moduli spaces of algebraic spaces of a given genus — coarse or fine, smooth or stable, possibly...
View ArticleDo these nonlocal gadgets define a strict hierarchy?
Definition: For $0\leq p\leq 1$, a $p$-trigadget is a pair of boxes, each with three input buttons (say, $X,Y,Z$, only one of which can be pressed) and an light which can flash one of two colors (blue...
View ArticleReference request: a real-valued semicontinuous function on a Baire space is...
I thought the following result was well-known, but I can't seem to find it in any standard textbook on real analysis or general topology:Theorem (★). Let $X$ be a Baire topological space and $f\colon...
View ArticleHow can we characterize cardinals of the form $\lambda^{\aleph_0}$?
This question is a followup to another one concerning the Hamel basis cardinalities (i.e., dimension qua vector space) of Hilbert spaces. Nik Weaver proved in an answer that they are exactly the...
View ArticleWhat does “the” mean in “the first Kleene algebra”? (In what sense is it...
Definition:“The” first Kleene algebra $\mathcal{K}_1$ is the set $\mathbb{N}$ of natural numbers endowed with the partial operation $(p,n) \mapsto p\bullet n := \varphi_p(n)$ where $\varphi_p$ is the...
View ArticleTopological spaces in which positive and negative values of a real function...
(This question is about classical mathematics, despite the appearance of constructive mathematics terms for reasons explained below.)Definition: Say that a topological space $X$satisfies LLPO when, for...
View ArticleDoes analytic WLLPO together with sequential LLPO imply analytic LLPO?
This question is about constructive mathematics, without any form of Choice except Unique Choice, such as in the internal logic of a topos with natural numbers object, or in IZF. The “reals” (and the...
View ArticleAnswer by Gro-Tsen for What to do if paper was rejected with good report?
This happened to me once, and the editor of the editor whose editorial board took the decision to reject the paper offered to disclose the referee's identity to another journal if I chose to resubmit...
View ArticleComment by Gro-Tsen on Is every real number the limit of a sequence of...
@MikhailKatz The problem is that, in constructive math in the absence of Choice, one cannot assert that every (Dedekind) real number is the limit of a sequence of rationals: those that are are the...
View ArticleComment by Gro-Tsen on Is every real number the limit of a sequence of...
Aw, you beat me to it. 😖
View ArticleComment by Gro-Tsen on Is every real number the limit of a sequence of...
@MohammadTahmasbizade The counterexample is easily modified to additionally satisfy $1\leq x\leq 2$, so it is not provable that a real in $[1,2]$ (or, a fortiori, a nonzero real) is the limit of a...
View ArticleAnswer by Gro-Tsen for Is every real number the limit of a sequence of...
This is more an extended comment than an answer (or at least it is a very partial answer, which I'm sure OP is already aware of), but because of the confusion such questions typically cause (as...
View ArticleComment by Gro-Tsen on What is known about the area of the symmetric...
I think the following extension of the computation should be doable with the same technique, and of some interest: let $a_n$ be the area that is covered by exactly $2^n$ squares (these are the only...
View ArticleComment by Gro-Tsen on About the operator $[x\ \int.dx]^n$
@JochenGlueck You're right! Everybody here knows that we shouldn't do this sort of things.
View ArticleAnswer by Gro-Tsen for Special rational numbers that appear as answers to...
According to this answer on this site, the area of the Pythagoras tree fractal (obtained by starting with a unit square, putting two smaller squares of size scaled down by $\sqrt{2}/2$ adjacent to one...
View ArticleComment by Gro-Tsen on Lengths of arithmetical sequences and arithmetical...
Don't you mean $S=\{n+bj : 0\leq j\leq k\}$ with $b\geq1$ perhaps? Otherwise it's strange to use the term “arithmetical” for what is just an interval of integers.
View ArticleComment by Gro-Tsen on Formalizations of The Matchstick Diagram...
@PaulCrowley Indeed, it doesn't seem to be in Gamow. On the other hand, I wrote this popular science text on ordinals for my high school's students' journal in 1998, where a very similar representation...
View ArticleComment by Gro-Tsen on Algorithm to decide whether two constructible numbers...
So, to make sure I understand correctly, you might represent $\sqrt{2-\sqrt{2}}$ either as $0 + 1\sqrt{2-\sqrt{2}} \in \mathbb{Q}[\sqrt{2}][\sqrt{2-\sqrt{2}}]$ or as $-\sqrt{2} +...
View ArticleWhy is it so hard to give examples of differentially closed fields?
The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability;...
View ArticleAnswer by Gro-Tsen for Open Sets in Hausdorff spaces
Here is a counterexample showing that it is not true in general that, for $X$ Hausdorff, every non-empty open set contains a closed set with non-empty interior.Let $X$ be the countable complement...
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