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Questions tagged [stone-cech-compactification]

21
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4answers
889 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 \...
14
votes
3answers
1k views

Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology

The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters. Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ ...
13
votes
1answer
284 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 ...
12
votes
0answers
291 views

Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$. Do the meromorphic functions separate the points of $S$? I ...
11
votes
0answers
647 views

A basic question on Stone-Cech compactification of $\mathbb{Z}$

Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...
8
votes
2answers
2k views

Stone-Čech compactification of $\mathbb R$

Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it ...
8
votes
0answers
113 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\...
8
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0answers
202 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 ...
8
votes
0answers
222 views

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 $...
6
votes
1answer
133 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 ...
6
votes
1answer
182 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 ...
6
votes
2answers
715 views

A question about the Stone–Čech compactification of discrete spaces

Let $D(\kappa)$ be the discrete space of cardinality $\kappa$, and $\beta D(\kappa)$ its Stone–Čech compactification. Is there, for every infinite cardinal $\kappa$, a subset $Y \in [\beta D(\kappa)]^...
6
votes
1answer
120 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-...
6
votes
1answer
201 views

Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
6
votes
1answer
420 views

zero-dimensional completely regular space with $\sigma$-complete clopen algebra

Suppose $X$ is a zero-dimensional completely regular space (clopen sets form a base) such that the Boolean algebra of clopen sets is a $\sigma$-complete Boolean algebra. Must $X$ be basically ...
6
votes
1answer
193 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 ...
5
votes
1answer
398 views

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 ...
5
votes
1answer
282 views

Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets?

Is the following statement true, and if it is, does someone have a reference? Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, =...
5
votes
0answers
259 views

Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
4
votes
1answer
323 views

What is the Stone–Čech compactification of a dense set of $\beta N \setminus N$?

Is the Stone–Čech compactification of a dense $G_\delta$-set $X \subset\beta N \setminus N$ homeomorphic to $\beta N \setminus N$? Here, $\beta N \setminus N$ is the complement of $N$ in the Stone–...
4
votes
1answer
145 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 ...
3
votes
1answer
234 views

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 ...
3
votes
1answer
105 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 ...
3
votes
2answers
260 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 ...
3
votes
0answers
99 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 ...
3
votes
0answers
58 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)...
2
votes
2answers
78 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 ...
2
votes
1answer
85 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 ...
2
votes
1answer
867 views

Invariant functionals on C(R) and amenable groups

Since there seems to be no progress in this interesting question, I took the liberty to reformulate it in a way, that is easier to understand. Moreover, my answer shows that the question is related to ...
2
votes
1answer
177 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 ...
2
votes
1answer
418 views

A completely regular space that is very non-normal

Take a completely regular Hausdorff topological space $X$ considered as a subset of its Stone-Čech compactification $\beta X$. If $X$ is not normal, we can find a closed subset $Y$ of $X$ and a ...
2
votes
0answers
261 views

Continuity of multiplicative character

Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
1
vote
1answer
132 views

Non-idempotent ultrafilters in the Stone-Cech compactification

Supposing that $\Gamma$ is an infinite, discrete group and that $\beta\Gamma$ is the Stone-Cech compactification of $\Gamma$, the group structure of $\Gamma$ can be extended to a semigroup structure ...
1
vote
0answers
225 views

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\...
1
vote
0answers
291 views

Type I subspaces of the Stone Cech compactification of $\omega$

EDIT: I found a construction, see below. I decided not to delete the question in case someone is interested. A space $X$ is of Type I if $X=\cup_{\alpha<\omega_1} X_\alpha$, where each $X_\alpha$ ...
0
votes
1answer
75 views

Is $C_b(Q,E)$ linearly isometric isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Stone-Cech 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 $Q$ while $C_b(Q,E)$ be the collection of all $E$-...
0
votes
2answers
175 views

Locally compact, 0-dimensional, pseudocompact space

Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional? It is known that a 0-dimensional locally compact space which ...
0
votes
1answer
542 views

Stone-Cech compatification and ultrafilter [closed]

I have been studding about compatification of a topological space $X$. But I have low understanding about the Stone-Cech compatification, specially construction of the Stone-Cech compatification on ...
0
votes
0answers
82 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 ...
-1
votes
1answer
183 views

Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem in Furstenberg's famous paper "non-commuting product." I try my best to understand Stone-Cech compactification theorem by ...