Questions tagged [stone-cech-compactification]

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Stone-Čech compactification

Is every hyperstonean space a Stone-Čech compactification of a discrete space? Is there a closed subset of Stone-Čech boundary that is extremally disconnected?
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Ultrafilters of closed sets

The following definition should be standard, but let me state it just in case there is some ambiguity: If $\mathscr{L}$ is a set of subsets of a set $X$ that is closed under finite unions and ...
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Is the Čech–Stone compactification of the integers always a retract of an extremally disconnected space?

Probably $\beta \mathbb N$ is not an absolute retract (is there an easy argument for this?), but I'd be interested to know what happens in the class of extremally disconnected (compact) spaces. Is it ...
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When are the zero sets of two continuous functions in the Stone-Čech compactification included in one another?

Let $X$ be a topological space (feel free to add some simplifying assumptions here, like “completely regular” provided at least the case of finite-dimensional manifolds is covered). Let $f,g \in C^*(...
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Is this space the Stone–Čech compactification?

Let $\mathbb{S}$ be the Sierpiński space, the two pointed space $\{ 0, 1 \}$ with open sets $\{0 \}$, $\emptyset$, $\{ 0, 1 \}$. We give $\{ 0, 1 \}$ a partial order where $0 < 1$. Let $X$ be a ...
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Are all free ultrafilters 'the same' in some sense?

Consider the set of ultrafilters $\beta(\mathbb N)$ on $\mathbb N$. Any function $f\colon\mathbb N\to\mathbb N$ extends to a function $\beta f\colon \beta \mathbb N \to \beta\mathbb N$. We say that ...
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Convergence properties in dense subsets of $\omega^*$

The space $\omega^*$, the remainder of the Cech-Stone compactification of the integers, fails to have all convergence-type properties known to me. Sequentiality. (As a matter of fact $\omega^*$ does ...
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Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?

Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as $$ \mathcal{U} * \mathcal{V} = \left\{ A \...
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Self-homeomorphism of Stone-Čech boundary with an isolated fixed point

$\DeclareMathOperator\bso{\beta^*\!\omega}\DeclareMathOperator\Homeo{Homeo}$Let $\bso$ be the complement of the countable discrete space $\omega$ in its Stone-Čech compactification $\beta\omega$ (some ...
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Which topoi are local with respect to Stone-Cech compactification?

Compact Hausdorff spaces $X$ are characterized among all topological spaces by the fact that for any topological space $S$, the embedding $S \to \beta S$ into its Stone-Cech compactification induces a ...
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The extension of the substitution map of the semigroup of variable words to its Stone–Čech compactification is a homomorphism

Reading the proof of the Hales-Jewett theorem the author defines $W_L$ as the set of finite words over some alphabet $L$, $W_{L_v}$ as the set of variable-words over $L$, i.e. finite words over $L \...
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Characterization of pretty compact spaces

This is a cross post from MSE. I believe that the following problem have already been considered by some sophisticated topologist. Definition 1. A non-compact Hausdorff topological space $X$ is called ...
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Stone-Čech boundary is not extremally disconnected

Recall that a topological space is called extremally disconnected if the closure of every open subset is still open. Every discrete space is of course extremally disconnected, and the standard non-...
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Minimal components of the translation action on the Stone–Čech compactification

$\newcommand\Cb{C^\text b}$Let $\Cb(\mathbb R)$ be the C*-algebra formed by all bounded, continuous, complex valued functions on $\mathbb R$. Consider the action $\tau $ of $\mathbb R$ on $\Cb(\...
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Addition and Rudin-Keisler ordering in $\beta \omega$

$\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that extends ...
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adding one point from the Stone-Cech compactification

Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification. The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...
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A permutation group inducing a topologically transitive action without dense orbits on $\omega^*$

Let $G$ be a subgroup of the permutation group $S_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech ...
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NCF, P-points, weak P-points, and cardinalities

The post is a bit long, but all the questions are similar or concern the same topic. Let $\omega^*=\beta\omega\setminus\omega$. A well-known topological definition of a P-point (on $\omega$) is as ...
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Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?

Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures. Consider the endomorphism $\hat{\Phi}$ ...
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12 votes
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Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?

Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder. The map $j:n\...
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3 votes
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Ultrafilter comonad on the category of Stone spaces

Let $\mathsf{Stone}$ denote the category of Stone spaces (compact, totally disconnected Hausdorff spaces) and continuous maps. The forgetful functor $U : \mathsf{Stone} \to \mathsf{Set}$ has a left ...
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Partitioning $\beta \mathbb{Z} \setminus \mathbb{Z}$

Take the integers $\mathbb{Z}$ and the addition \begin{align*} +: \mathbb{Z} \times \mathbb{Z} &\to \mathbb{Z} \\ (a,b) &\mapsto a+b. \end{align*} Using the Stone-Čech compactification $...
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The Stone-Čech compactification of a inverse system

Is the Stone-Čech compactification of the inverse limit of an inverse system $\left\{ X_{i},f_{ij},I\right\} $ of Tychonoff spaces equal to the limit of the inverse system $\left\{ \beta X_{i},\beta ...
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16 votes
1 answer
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Best introductory texts on pointless topology

As I understand it, there are three canonical textbooks on pointless topology: the classic "Stone Spaces" by Johnstone, "Topology via Logic" by Steve Vickers, and the newer "Frames and Locales" by ...
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Stone duality- a modification

Let $2$ be the discrete topological space with two elements. For a map of sets $$\beta : X \times Y \rightarrow 2 $$ We get a topology on $X$ and a topology on $Y$. The topology on $X$ is the weakest ...
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Functor from rings into compact Hausdorff spaces

There is an adjunction $\text{Bool}^{op} \leftrightarrow \text{Set}$ between boolean algebras and sets which sends a boolean algebra to the set of its prime ideals and a set $X$ to $[X, \mathbb{F}_2]_{...
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Does surjective map induce surjective map on Hewitt real compactifications?

Let $\beta X$ be the Stone-Čech compactification and $\upsilon X$ be the Hewitt real compactification of a completely regular space $X$. It is well known that any continuous surjective map $f:X\...
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Tychonoff-ization and Urysohn (functionally Hausdorff) topological spaces

Let me first make sure I have the correct definitions because my question will be about the difference about the two and there may be some massive confusion on my part. A topological space $X$ is ...
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6 votes
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When Stone–Čech compactification is totally disconnected

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets, and a topological space, $X$ is called completely regular exactly in case points can be ...
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Continuous binary operations on $\beta\mathbb{N}$

It is well-known that the operation of addition of two ultrafilters on the set $\mathbb{N}$ of natural numbers which extends the natural addition on $\mathbb{N}$ to $\beta\mathbb{N}$, the Cech-Stone ...
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7 votes
1 answer
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Embeddability into $\beta\omega$ and $\omega^*$

It is well known that under CH every totally-disconnected compact F-space of weight at most $\omega_1$ can be embedded into the remainder $\omega^*=\beta\omega\setminus\omega$ of the Cech-Stone ...
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2 votes
1 answer
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Stone-Cech Compactification of the real line

I have a question in $\beta\mathbb{R}$, the Stone-Cech compactification of the real line $\mathbb{R}$. My question is: is $\beta\mathbb{R}$ a $\mathrm{F}$-space, i.e., the closure of two disjoint open ...
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6 votes
1 answer
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The Stone-Čech compactification of the fixed point set

Let $G$ be a discrete group and $X$ be a Tychonoff $G$-space. Then there exists a $G$-action on Stone-Čech compactification $\beta X$. If the fixed point set $X^{G}\neq \emptyset $, then the Stone-...
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3 votes
1 answer
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Borel $\sigma$-algebra in $\beta \mathbb N \times \beta \mathbb N$

For a second-countable space $X$, we have $${\rm Bor}(X\times X) = {\rm Bor}\, X \otimes {\rm Bor}\, X,$$ that is the Borel $\sigma$-algebra of the product is the product $\sigma$-algebra. Some ...
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8 votes
0 answers
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Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$

Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...
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2 votes
1 answer
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Are separability and ccc equivalent for closed subspaces of $\beta N$?

Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable ...
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6 votes
1 answer
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Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?

Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a ...
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Is $C_b(Q,E)$ linearly isometrically isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Stone–Čech compactification of $Q$?

Let $Q$ be a locally compact Hausdorff space and $E$ be a Banach space. Let $C(Q)$ be the collection of all real-valued continuous functions on $Q$ and $C_b(Q,E)$ be the collection of all $E$-valued ...
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What are the components of the Stone-Cech Remainder?

Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a ...
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Does any subset of $\beta\omega$ of cardinality $\mathfrak{c}$ have a weak P-point in its closure?

I believe that anyone who can answer this question knows the terminology, but $\beta\omega$ is the Čech-Stone compactification of the integers and a point $p$ is a weak P-point in a space $X$ iff it ...
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3 votes
0 answers
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On the compactification of partial semigroups

We begin by introducing some relevant definitions. Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)...
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15 votes
1 answer
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Continuous images of $\beta \mathbb{N} \setminus\mathbb{N}$

Let $\beta \mathbb{N}$ denote the Stone-Cech compatification of the natural numbers and $\beta \mathbb{N} \setminus\mathbb{N}$ denote the reminder of this compactification. I wonder if there is a ...
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6 votes
1 answer
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Is there a compactification with nontrivial connected remainder?

Question: Let $X$ be a continuum and $p \in X$. Under what conditions does there exist a compactification $\gamma (X-p)$ with $\gamma (X-p) - (X-p)$ connected and nondegenerate? Throughout, $X$ is a ...
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Is there a 'local' version of Near Coherence of Filters?

The axiom Near Coherence of Filters (NCF) is known to be independent of ZFC. Axiom (NCF I): For any two free ultrafilters $\mathcal D$ and $\mathcal E$ on $\mathbb N$, there exist finite-to-one ...
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Can we express separability of a ray-remainder in terms of the function algebra?

Let $X = [0, 1)$ be a ray and $C(X)$ the algebra of bounded continuous real functions. The spectrum of $C(X)$ is the Stone-Cech compactification $\beta [0,1) $ of the ray. It's easy to see the ...
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23 votes
4 answers
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Is $\beta \mathbb{N}$ homeomorphic to its own square?

Let $\mathbb{N}$ be the set of natural numbers and $\beta \mathbb N$ denotes the Stone-Cech compactification of $\mathbb N$. Is it then true that $\beta \mathbb N\cong \beta \mathbb N \times \beta \...
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6 votes
1 answer
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The need for nets in topology

I remember when I first heard about nets in topology (called also Moore-Smith sequences). I was told that most of useful topological properties which can be exressed in terms of sequences in the ...
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3 votes
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Sigma algebras on the Stone–Čech compactification of a countable discrete group

Let $\Gamma$ be a countable discrete group and $\beta \Gamma$ be its Stone–Čech compactification. My question is that Does the $\sigma$-algebra generated by clopen sets in $\beta \Gamma$ equal to ...
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Remote points in $\beta X$

It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space $...
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Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions

I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to $\beta\...
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