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Results tagged with stone-cech-compactification
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user 22277
17
votes
Accepted
Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology
We can also describe $\beta(\mathbb{Z},\mathcal{T})$ in terms of ultrafilters on Boolean algebras. I claim that $\beta(\mathbb{Z},\mathcal{T})$ is the space of ultrafilters on the Boolean algebra of c …
11
votes
Non-trivial convergent sequence in Stone-Čech compactification of $\mathbb{N}$
Let me give a fairly direct proof. Assume $R=(\mathcal{U}_{n})_{n}$ is a sequence of distinct ultrafilters on some set $X$. Since every infinite Hausdorff space has an infinite discrete subspace, ther …
9
votes
Accepted
Sigma algebras on the Stone–Čech compactification of a countable discrete group
No.
First of all, take note that for compact zero-dimensional spaces $X$, the $\sigma$-algebra generated by all clopen sets is precisely the $\sigma$-algebra of Baire sets (recall that in a completel …
7
votes
Accepted
When are the zero sets of two continuous functions in the Stone-Čech compactification includ...
Lemma: Suppose that $X$ is a compact Hausdorff space. Let $f,g:X\rightarrow[0,\infty)$ be continuous functions. Then the following are equivalent:
$Z(f)\subseteq Z(g)$.
For all $\epsilon>0$, there e …
7
votes
Non-trivial convergent sequence in Stone-Čech compactification of $\mathbb{N}$
This result follows from much stronger results from general topology. These results can be found in [1].
$\mathbf{Theorem}$ Each non-discrete closed subset of $\beta X\setminus\upsilon X$
conta …