All Questions
Tagged with stone-cech or stone-cech-compactification
86 questions
0
votes
1
answer
99
views
A question about G-Hewitt spaces
In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
- 1,367
1
vote
2
answers
202
views
Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact
Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications:
Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
- 1,201
11
votes
2
answers
314
views
Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem
Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent:
Every compactification of $X$ is zero-dimensional....
- 1,201
4
votes
1
answer
136
views
$\bf2$-Stone-Čech compactification of a product of topological spaces
Let $\beta_{\bf2} S$ be a compact, totally-disconnected space containing a dense, discrete subspace $S$ such that any function $f:S\to\bf2$ extends to a continuous map $\hat f:\beta_{\bf2} S\to\bf2$, ...
- 1,644
3
votes
1
answer
161
views
Stone-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 $\land$, $\lor$ ...
- 32.4k
1
vote
0
answers
101
views
When is the "Gelfand Remainder" compact?
Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
- 1,955
1
vote
1
answer
152
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 ...
0
votes
1
answer
109
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 ...
- 1,644
4
votes
1
answer
169
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 ...
- 1,367
3
votes
1
answer
131
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 ...
0
votes
1
answer
107
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\...
- 1,367
0
votes
1
answer
96
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.
- 1,367
22
votes
1
answer
711
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 ...
- 1,201
1
vote
1
answer
165
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 ...
- 151
1
vote
1
answer
360
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 ...
- 1,367
12
votes
1
answer
624
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
372
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?
- 149
7
votes
1
answer
322
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 ...
- 32.4k
0
votes
1
answer
199
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 ...
- 11.3k
8
votes
2
answers
398
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^*(...
- 32.4k
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 ...
user30211
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 ...
- 974
9
votes
2
answers
451
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 ...
- 4,413
1
vote
0
answers
121
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 \...
- 59
5
votes
2
answers
314
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 ...
- 63.9k
14
votes
0
answers
288
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 ...
- 63.9k
2
votes
1
answer
147
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 \...
- 59
8
votes
1
answer
272
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 ...
- 1,697
8
votes
4
answers
526
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-...
- 2,998
2
votes
1
answer
187
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(\...
- 483
4
votes
1
answer
222
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 ...
- 48.8k
7
votes
0
answers
221
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}$-...
- 71
1
vote
1
answer
136
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 ...
- 41.8k
5
votes
1
answer
362
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 ...
- 1,151
10
votes
1
answer
354
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}$ ...
- 63.9k
14
votes
2
answers
502
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\...
- 63.9k
3
votes
2
answers
416
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 ...
- 63.1k
7
votes
1
answer
385
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 $...
- 568
1
vote
1
answer
215
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 ...
- 1,367
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 ...
user147820
1
vote
0
answers
117
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 ...
user30211
4
votes
1
answer
386
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]_{...
user30211
0
votes
1
answer
92
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\...
- 1,367
6
votes
1
answer
420
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 ...
- 32.4k
6
votes
1
answer
582
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 ...
- 61
7
votes
1
answer
224
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 ...
- 1,151
7
votes
1
answer
230
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 ...
- 1,151
2
votes
1
answer
506
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 ...
- 21
6
votes
1
answer
207
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-...
- 1,367
3
votes
1
answer
735
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 ...
- 105