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 What makes the surreals special among other...
@JoelDavidHamkins Indeed, but all the “surreal-like” fields I mention in this question are isomorphic as ordered fields, so they all satisfy this property. The extra datum I am trying to grasp is the...
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 Stone-Čech boundary is not extremally disconnected
The question has already been answered satisfactorily, but I think the following presentation, which is implicit in the second part of YCor's answer, will clarify things. The fact that $\beta\mathbb{N}...
View ArticleAnswer by Gro-Tsen for Clarification on proof of the algebraic completeness...
[Converted from a comment into an answer, and expanded with a copy of the statement from Siegel's book.]You may find clearer the proof given in Siegel, Combinatorial Game Theory (2013, AMS Graduate...
View ArticleStone-Čech compactification of a Boolean subalgebra of $\{0,1\}^S$
Setup: Let $S$ be a set. Let $B$ be a Boolean subalgebra of $\{0,1\}^S$; i.e., just to be clear $B$ contains the constant $0$ and $1$ functions, and is stable under binary pointwise $\land$, $\lor$ and...
View ArticleComment by Gro-Tsen on Is there a mathematical theory of negotiation games?
@StevenLandsburg Thanks! Formula 7.4.13 in the chapter you cite gives $x=(c+d_1-d_2)/2$ when $F(x)=(x-d_1)(c-x-d_2)$, confirming that Nash's result subsumes the particular case I discussed in ¶5. If...
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 ArticleComment by Gro-Tsen on Removing disks with maximal radius in a disk: must the...
I think your idea works, but it's a bit more complicated than you describe: if you just punch the $n$ circles you said in each strip, the remainder won't be nowhere dense as required. I'm fairly...
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 ArticleComment by Gro-Tsen on Intersecting a smooth hypersurface with planes
@LorenzoAndreaus I think the results of SGA 7 exp. XVII (“Pinceaux de Lefschetz: théorème d'existence” by Katz), like thm. 2.5 and prop. 3.2 should answer your question. (Available online around here.)
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 ArticleComment by Gro-Tsen on Is there a topological space $X$ such that every...
① This sort of question should be posted on Math StackExchange, not MathOverflow. ② It's already answered here on Math Stackexchange (for “regular open” instead of “regular closed” but of course this...
View ArticleComment by Gro-Tsen on Cardinals arising as the real vector space dimension...
The $\lambda^{\aleph_0}$ are cardinals of uncountable cofinality which are $\geq 2^{\aleph_0}$. But are they all the cardinals of uncountable cofinality which are $\geq 2^{\aleph_0}$? I think this need...
View ArticleComment by Gro-Tsen on Points of continuity of a lower semicontinuous...
If instead of a condition on $f$ we ask for a condition on $X$, one such is discussed in this related question (there called the “super Baire” property).
View ArticleComment by Gro-Tsen on Reference request: a real-valued semicontinuous...
Sorry, maybe I wasn't clear: I wasn't asking for a proof, I have a proof, I was asking for a citeable reference.
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 ArticleComment by Gro-Tsen on Does the decomposability of $\mathbb{R} \setminus...
To clarify the question, by $\mathbb{R}\setminus\mathbb{Q}$ you mean $\{x\in\mathbb{R}:\forall r\in\mathbb{Q}.\neg(x=r)\}$ rather than $\{x\in\mathbb{R}:\forall r\in\mathbb{Q}.(x\#r)\}$, correct? (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...
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