Post History
#3: Post edited
by
Michael Hardy
·
2026-06-14T01:21:35Z (3 months ago)
The code \mathcal{CB} was not between "dollar signs" and so not rendered. Using \prod rather than \Pi results in correct formatting of the subscript and other differences.
Stone–Čech compactification and ultrafilters
Munkres defines the Stone–Čech compactification $\beta X$ of a topological space $X$ as based on the embedding of completely regular spaces into the cube $[0,1]^{\mathcal{CB}(X)}$ where I use \mathcal{CB}(X)} to denote the space of bounded continuous functions on $X$. Specifically, any point $h$ in this cube is defined as an assignment of values $f(h)$ for each bounded continuous function $f: X → \mathbb{R}$; and then $\beta X$ is the closure of $X$ in the aforesaid cube.[^1]- (More explicitly, we'd write the cube $\mathfrak{C}$ as)
\[\beta X \subseteq \mathfrak{C} := \Pi_{f\in\mathcal{CB}(X)} [\text{glb}(f),\text{lub}(f)]\]- (and thus any point $x \in X$ would embed to the point $\{f(x)\}_{f\in\mathcal{CB}(X)}$ in $\mathfrak{C}$.)
- This seems like a fairly straightforward definition based on continuous functions. I read in Wikipedia, though, that there is a nontrivially different characterization of the Stone–Čech compactification based on ultrafilters![^3]
- Is this correct? If so, approximately why is it correct?
- ---
- [^1]: For vague intuition, I think we could say that $\beta X$ is almost like a "dual space of the dual space" of $X$.
- [^3]: E.g. in Wikipedia's article on the [Wallman compactification](https://en.wikipedia.org/wiki/Wallman_compactification), which despite being defined based on ultrafilters and not based on bounded continuous functions, is apparently "essentially the same as the Stone–Čech compactification." So I'm wondering why it is that the Wallman and Stone–Čech compactifications would coincide (if indeed they do) on normal spaces.
- Munkres defines the Stone–Čech compactification $\beta X$ of a topological space $X$ as based on the embedding of completely regular spaces into the cube $[0,1]^{\mathcal{CB}(X)}$ where I use $\mathcal{CB}(X)$ to denote the space of bounded continuous functions on $X$. Specifically, any point $h$ in this cube is defined as an assignment of values $f(h)$ for each bounded continuous function $f: X → \mathbb{R}$; and then $\beta X$ is the closure of $X$ in the aforesaid cube.[^1]
- (More explicitly, we'd write the cube $\mathfrak{C}$ as)
- \[\beta X \subseteq \mathfrak{C} := \prod_{f\in\mathcal{CB}(X)} [\text{glb}(f),\text{lub}(f)]\]
- (and thus any point $x \in X$ would embed to the point $\{f(x)\}_{f\in\mathcal{CB}(X)}$ in $\mathfrak{C}$.)
- This seems like a fairly straightforward definition based on continuous functions. I read in Wikipedia, though, that there is a nontrivially different characterization of the Stone–Čech compactification based on ultrafilters![^3]
- Is this correct? If so, approximately why is it correct?
- ---
- [^1]: For vague intuition, I think we could say that $\beta X$ is almost like a "dual space of the dual space" of $X$.
- [^3]: E.g. in Wikipedia's article on the [Wallman compactification](https://en.wikipedia.org/wiki/Wallman_compactification), which despite being defined based on ultrafilters and not based on bounded continuous functions, is apparently "essentially the same as the Stone–Čech compactification." So I'm wondering why it is that the Wallman and Stone–Čech compactifications would coincide (if indeed they do) on normal spaces.
#2: Post edited
by
clemens
·
2026-05-10T13:11:22Z (4 months ago)
added expression for the cube
Munkres defines the Stone–Čech compactification $\beta X$ of a topological space $X$ as based on the embedding of completely regular spaces into the cube $[0,1]^{\mathcal{CB}(X)}$ where I use \mathcal{CB}(X)} to denote the space of bounded continuous functions on $X$. Specifically, any point $h$ in this cube is defined as an assignment of values $f(h)$ for each bounded continuous function $f: X → [0,1]$; and then $\beta X$ is the closure of $X$ in the aforesaid cube.[^1]This seems like a fairly straightforward definition based on continuous functions. I read in Wikipedia, though, that there is a nontrivially different characterization of the Stone–Čech compactification based on ultrafilters![^2]- Is this correct? If so, approximately why is it correct?
- ---
- [^1]: For vague intuition, I think we could say that $\beta X$ is almost like a "dual space of the dual space" of $X$.
[^2]: E.g. in Wikipedia's article on the [Wallman compactification](https://en.wikipedia.org/wiki/Wallman_compactification), which despite being defined based on ultrafilters and not based on bounded continuous functions, is apparently "essentially the same as the Stone–Čech compactification." So I'm wondering why it is that the Wallman and Stone–Čech compactifications would coincide (if indeed they do) on normal spaces.
- Munkres defines the Stone–Čech compactification $\beta X$ of a topological space $X$ as based on the embedding of completely regular spaces into the cube $[0,1]^{\mathcal{CB}(X)}$ where I use \mathcal{CB}(X)} to denote the space of bounded continuous functions on $X$. Specifically, any point $h$ in this cube is defined as an assignment of values $f(h)$ for each bounded continuous function $f: X → \mathbb{R}$; and then $\beta X$ is the closure of $X$ in the aforesaid cube.[^1]
- (More explicitly, we'd write the cube $\mathfrak{C}$ as)
- \[\beta X \subseteq \mathfrak{C} := \Pi_{f\in\mathcal{CB}(X)} [\text{glb}(f),\text{lub}(f)]\]
- (and thus any point $x \in X$ would embed to the point $\{f(x)\}_{f\in\mathcal{CB}(X)}$ in $\mathfrak{C}$.)
- This seems like a fairly straightforward definition based on continuous functions. I read in Wikipedia, though, that there is a nontrivially different characterization of the Stone–Čech compactification based on ultrafilters![^3]
- Is this correct? If so, approximately why is it correct?
- ---
- [^1]: For vague intuition, I think we could say that $\beta X$ is almost like a "dual space of the dual space" of $X$.
- [^3]: E.g. in Wikipedia's article on the [Wallman compactification](https://en.wikipedia.org/wiki/Wallman_compactification), which despite being defined based on ultrafilters and not based on bounded continuous functions, is apparently "essentially the same as the Stone–Čech compactification." So I'm wondering why it is that the Wallman and Stone–Čech compactifications would coincide (if indeed they do) on normal spaces.
#1: Initial revision
by
clemens
·
2026-05-09T21:01:49Z (4 months ago)
Stone–Čech compactification and ultrafilters
Munkres defines the Stone–Čech compactification $\beta X$ of a topological space $X$ as based on the embedding of completely regular spaces into the cube $[0,1]^{\mathcal{CB}(X)}$ where I use \mathcal{CB}(X)} to denote the space of bounded continuous functions on $X$. Specifically, any point $h$ in this cube is defined as an assignment of values $f(h)$ for each bounded continuous function $f: X → [0,1]$; and then $\beta X$ is the closure of $X$ in the aforesaid cube.[^1]
This seems like a fairly straightforward definition based on continuous functions. I read in Wikipedia, though, that there is a nontrivially different characterization of the Stone–Čech compactification based on ultrafilters![^2]
Is this correct? If so, approximately why is it correct?
---
[^1]: For vague intuition, I think we could say that $\beta X$ is almost like a "dual space of the dual space" of $X$.
[^2]: E.g. in Wikipedia's article on the [Wallman compactification](https://en.wikipedia.org/wiki/Wallman_compactification), which despite being defined based on ultrafilters and not based on bounded continuous functions, is apparently "essentially the same as the Stone–Čech compactification." So I'm wondering why it is that the Wallman and Stone–Čech compactifications would coincide (if indeed they do) on normal spaces.