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Results tagged with stone-cech-compactification
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user 14094
6
votes
Non-trivial convergent sequence in Stone-Čech compactification of $\mathbb{N}$
Here's an answer for sequences of arbitrary ultrafilters (I replace $\mathbf{N}$ by an arbitrary set $X$ since it holds in general). Let $\mathcal{U}_n$ be a sequence of ultrafilters converging to an …
- 63.9k
11
votes
Is $\beta \mathbb{N}$ homeomorphic to its own square?
The negative answer is equivalent to showing that there are two disjoint subsets $A,B$ of $\mathbf{N}^2$ with non-disjoint closures in $(\beta\mathbf{N})^2$. This be made explicit: take $A=\{(n,m):n=m …
- 63.9k
6
votes
Accepted
Stone-Čech boundary is not extremally disconnected
We can suppose $X=\omega$. Let $(X_i)_{i\in I}$ be a continuum family of infinite subsets of $\omega$ with pairwise finite intersection. Define $Y_i=\bar{X_i}-X_i$. So the $Y_i$ are pairwise disjoint …
- 63.9k