# Questions tagged [stone-cech-compactification]

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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 $\beta ... 0 votes 1 answer 91 views ### Extending maps from a discrete set to a Stone-Čech compactification while retaining an injectivity condition For$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)$has the ... • 1,388 4 votes 1 answer 153 views ### Stone–Čech compactification and an ultrafilter of regular closed sets$\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$. The family of all regular closed sets of a ... • 1,211 3 votes 1 answer 125 views ### Spectrum of continuous functions as a semigroup Let$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)$, namely the ... 0 votes 1 answer 93 views ### A question about the Stone-Čech compactification and ultrafilter Let$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\...
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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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### 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 ...