Questions tagged [stone-cech-compactification]
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73
questions
19
votes
1
answer
505
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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 ...
1
vote
1
answer
122
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 ...
1
vote
1
answer
312
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 ...
11
votes
1
answer
510
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 ...
5
votes
1
answer
334
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?
5
votes
1
answer
238
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 ...
0
votes
1
answer
120
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 ...
8
votes
2
answers
380
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^*(...
8
votes
3
answers
1k
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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 ...
30
votes
3
answers
2k
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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 ...
8
votes
2
answers
381
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 ...
1
vote
0
answers
112
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 \...
5
votes
2
answers
266
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 ...
13
votes
0
answers
259
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 ...
2
votes
1
answer
141
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 \...
8
votes
1
answer
239
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 ...
3
votes
1
answer
244
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-...
2
votes
1
answer
164
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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(\...
4
votes
1
answer
213
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 ...
7
votes
0
answers
202
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}$-...
1
vote
1
answer
121
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 ...
5
votes
1
answer
293
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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 ...
8
votes
0
answers
266
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}$ ...
12
votes
1
answer
381
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\...
3
votes
2
answers
363
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 ...
7
votes
1
answer
372
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 $...
1
vote
1
answer
190
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 ...
16
votes
1
answer
1k
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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 ...
1
vote
0
answers
110
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 ...
4
votes
1
answer
342
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]_{...
0
votes
1
answer
79
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\...
6
votes
1
answer
354
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 ...
6
votes
1
answer
487
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 ...
7
votes
1
answer
214
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 ...
7
votes
1
answer
219
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 ...
2
votes
1
answer
418
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 ...
6
votes
1
answer
178
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-...
3
votes
1
answer
582
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 ...
8
votes
0
answers
153
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\...
2
votes
1
answer
134
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 ...
6
votes
1
answer
324
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 ...
1
vote
2
answers
185
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 ...
3
votes
2
answers
482
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 ...
2
votes
2
answers
154
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 ...
3
votes
0
answers
72
views
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)...
15
votes
1
answer
481
views
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 ...
6
votes
1
answer
272
views
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 ...
8
votes
0
answers
219
views
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 ...
0
votes
0
answers
89
views
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 ...
23
votes
4
answers
1k
views
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 \...