Skip to main content

Questions tagged [stone-cech-compactification]

Filter by
Sorted by
Tagged with
1 vote
1 answer
130 views

Points in the Stone Cech compactification are intersection of open sets

Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
Serge the Toaster's user avatar
0 votes
1 answer
91 views

Extending maps from a discrete set to a Stone-Čech compactification while retaining an injectivity condition

For $S$ a set, let $\beta_{\bf2}(S)$ be a compact, totally disconnected space containing $S$ where $S$ in the subspace topology is discrete and $S$ is a dense subspace, and $\beta_{\bf2}(S)$ has the ...
Tri's user avatar
  • 1,388
4 votes
1 answer
153 views

Stone–Čech compactification and an ultrafilter of regular closed sets

$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$A subset $A$ of a topological space $X$ is called regular closed if $A=\cl _{X}\int_{X}A$. The family of all regular closed sets of a ...
Mehmet Onat's user avatar
  • 1,211
3 votes
1 answer
125 views

Spectrum of continuous functions as a semigroup

Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the ...
Serge the Toaster's user avatar
0 votes
1 answer
93 views

A question about the Stone-Čech compactification and ultrafilter

Let $X$ be a Tychonoff space and let $\beta X$ is the Stone-Čech compactification of $X$. Assume $f:X\longrightarrow \mathbb{R}$ is a bounded function. Then there exists a function $f^{\beta }:\beta X\...
Mehmet Onat's user avatar
  • 1,211
0 votes
1 answer
95 views

A question about filterbasis

K. Hardy and R. G. Wood assert that the family in line 4 is a filterbase. I couldn't show it.
Mehmet Onat's user avatar
  • 1,211
22 votes
1 answer
616 views

Are $\beta \mathbb{Q}$ and $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ homeomorphic?

The canonical inclusion $\beta\mathbb{Q}\setminus \mathbb{Q} \hookrightarrow \beta\mathbb{Q}$ is not the Stone-Čech compactification of $\beta\mathbb{Q}\setminus \mathbb{Q}$. Even so, this doesn't ...
Jakobian's user avatar
  • 817
1 vote
1 answer
152 views

Trivial convergent sequences in $\beta X$

Let $X$ be a Tychonoff space and denote by $\beta X$ its Stone-Čech compactification. We know, for example, that if $X$ is an $F$-space then $\beta X$ is an $F$-space and, therefore, in $\beta X$, the ...
Carlos Jiménez's user avatar
1 vote
1 answer
340 views

A question about realcompact spaces

Let $X$ be completely regular space, $\beta X$ be Stone-Čech compactification of $X$, and $\upsilon X$ be Hewitt realcompactification of $X$. Then $X\subset \upsilon X\subset \beta X$. If the ...
Mehmet Onat's user avatar
  • 1,211
12 votes
1 answer
595 views

Stone–Čech compactification as a semigroup

Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
Serge the Toaster's user avatar
5 votes
1 answer
363 views

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?
Sherlok's user avatar
  • 149
7 votes
1 answer
305 views

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 ...
Gro-Tsen's user avatar
  • 30.3k
0 votes
1 answer
176 views

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 ...
Tomasz Kania's user avatar
  • 11.3k
8 votes
2 answers
388 views

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^*(...
Gro-Tsen's user avatar
  • 30.3k
8 votes
3 answers
1k views

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 ...
user avatar
31 votes
3 answers
2k views

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 ...
Squala's user avatar
  • 974
9 votes
2 answers
432 views

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 ...
Santi Spadaro's user avatar
1 vote
0 answers
120 views

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 \...
andpe's user avatar
  • 59
5 votes
2 answers
302 views

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 ...
YCor's user avatar
  • 60.8k
13 votes
0 answers
275 views

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 ...
Tim Campion's user avatar
  • 61.9k
2 votes
1 answer
146 views

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 \...
andpe's user avatar
  • 59
8 votes
1 answer
259 views

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 ...
Norbert's user avatar
  • 1,687
6 votes
2 answers
360 views

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-...
AlexE's user avatar
  • 2,946
2 votes
1 answer
178 views

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(\...
Black's user avatar
  • 471
4 votes
1 answer
221 views

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 ...
Dominic van der Zypen's user avatar
7 votes
0 answers
218 views

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 $G_{\delta}$-...
Arkady's user avatar
  • 71
1 vote
1 answer
134 views

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 ...
Taras Banakh's user avatar
5 votes
1 answer
341 views

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 ...
Damian Sobota's user avatar
10 votes
1 answer
345 views

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 $\hat{\Phi}$ ...
YCor's user avatar
  • 60.8k
14 votes
2 answers
484 views

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\...
YCor's user avatar
  • 60.8k
3 votes
2 answers
386 views

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 ...
Martin Brandenburg's user avatar
7 votes
1 answer
380 views

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 $...
André Caldas's user avatar
1 vote
1 answer
206 views

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 ...
Mehmet Onat's user avatar
  • 1,211
17 votes
1 answer
2k views

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 ...
user avatar
1 vote
0 answers
116 views

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 ...
user avatar
4 votes
1 answer
366 views

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]_{...
user avatar
0 votes
1 answer
88 views

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\...
Mehmet Onat's user avatar
  • 1,211
6 votes
1 answer
394 views

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 ...
Gro-Tsen's user avatar
  • 30.3k
6 votes
1 answer
547 views

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 ...
Arena's user avatar
  • 61
7 votes
1 answer
219 views

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 ...
Damian Sobota's user avatar
7 votes
1 answer
225 views

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 ...
Damian Sobota's user avatar
2 votes
1 answer
477 views

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 ...
user132068's user avatar
6 votes
1 answer
199 views

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-...
Mehmet Onat's user avatar
  • 1,211
3 votes
1 answer
693 views

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 ...
user3522356's user avatar
8 votes
0 answers
159 views

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\...
Lviv Scottish Book's user avatar
2 votes
1 answer
143 views

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 ...
user129208's user avatar
6 votes
1 answer
337 views

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 ...
Taras Banakh's user avatar
1 vote
2 answers
207 views

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 ...
Idonknow's user avatar
  • 603
3 votes
2 answers
522 views

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 ...
Daron's user avatar
  • 1,761
2 votes
2 answers
171 views

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 ...
Mathieu Baillif's user avatar