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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 …
Martin Sleziak's user avatar
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 …
YCor's user avatar
  • 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 …
YCor's user avatar
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