# Questions tagged [stone-cech-compactification]

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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\...

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### 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.

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...