# Tagged Questions

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### Stone-Čech compactification using ultrafilters

Let $X = \omega \cup \{ x \}$ ne the Stone-Čech compactification of $\omega$. (I am viewing $X$ as a subspace of the set of ultrafilters over $\omega$). Let, $\mathcal A$, $\mathcal B$ be two disjoint ...
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### A boolean algebra is complete if its stone space is extremally disconnected

I have the following proof, but I don't understand one of the steps: Theorem 4.4. A Boolean algebra is complete iff its Stone space is exlremally disconnected. Proof. Identify the given ...
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### Čech-Stone compactification of $\mathbb N$ and ultrafilters on $\mathbb N$

I have found in the literature that the Čech-Stone compactification $\beta\mathbb N$ of $\mathbb N$ (or more generally, of any discrete topological space) can be identified with ultrafilters on ...
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### What does it mean for an ultrafilter to have a limit?

I got this question from the construction of the Stone-Čech compactification using ultrafilters given in Wikipedia. There they say that if $F$ is an ultrafilter in a compact Hausdorff space $K$ then ...
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### the Stone-Čech compactification by using ultrafilters

If $X$ is discrete, one can construct $\beta X$ as the set of all ultrafilters on $X$. But which kind of topology must we use in the above sentence? How can we define the Stone–Čech ...
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### Stone-Čech compactification of a discrete space

I would like to know: Why is the Stone-Čech compactification of a discrete space the set of ultrafilters on that space?
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### Every minimal KC space is compact

A space $(X,\tau )$ is said to be minimal $KC$ if $(X,\tau)$ is $KC$ but no topology on $X$ which is strictly smaller than $\tau$ is $KC$. Theorem : Every minimal KC space is compact. Proof. ...
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### topological space

Let $( X,\tau )$ be a $T_1$ topological space. Let $D = \{ d_n : n \in \omega \}$ be a countably infinite closed discrete subspace of $X$. Fix $P \in X$ and let $F \in \beta\omega- \omega$ be an ...
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### Why does the z-ultrafilter Stone-Čech compactification construction have the universal property?

For the notions of $z$-filter, prime $z$-filters and $z$-ultrafilters see A Prime $\mathcal P$-filter is contained in a unique $\mathcal P$-ultrafilter? Let $X$ be a Tychonoff space. Let $BX$ be the ...
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### Another question in relation to Tychonoff theorem

Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
I want to show for the collection of ultra filters on a (non-empty) set $A$, $U(A)$. That $U(A)$ is compact where the topology is derived from the base $U_B = \{F\in U(A)|B\in F\}$. Seeing as $A$ can ...