All Questions
Tagged with compactification set-theory
5 questions
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What is the cardinality of the inverse image of a point under the map $β(ℕ^2)→β(ℕ)^2$?
I will write $β$ for the Stone-Čech compactification. Let $p$ be the canonical map $β(ℕ^2)→β(ℕ)^2$. Let $(u,v) ∈ β(ℕ)^2$. If $u$ or $v$ is in $ℕ$, then $p^{-1}(u,v)$ is a singleton. Otherwise, $p^{-1}(...
- 2,091
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Why does the proper class construction of Stone–Čech compactification fail?
Wikipedia's article on the Stone–Čech compactification gives several constructions of it, one which is this:
One attempt to construct the Stone–Čech compactification of $X$ is to take the closure of ...
- 10.6k
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Why is an infinite compact Hausdorff space with a dense countable subset a compactification of $Z^+$?
Studying General Topology from Munkres, I just read about Stone-Čech compactifications for the first time, tried to solve the exercises provided by the author and I came across the following one:
"...
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A well-behaved subset of $\beta\mathbb{R}$
Let $\beta X$ denote the usual Stone-Čech compactification of a Tychonoff space $X$. Let $S \subset \beta\mathbb{R}$ denote those points in $\beta \mathbb{R}$ which can be obtained as a limit point of ...
- 587
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Remote points in $\beta X$
It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space $...
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