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I meet the space $X$ of ultrafilters on $N$ with the topology generated by sets of the form $\{p\}\cup A$ where $A\in p \in X$. I can't understand the definition of the topology. Is the points in $N$ are all discrete? Could someone help me to understand this space? Any help will be appreciated:)

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2 Answers 2

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The points of $\Bbb N$ are isolated, so $\Bbb N$ forms a discrete open subset of $X$. $X\setminus\Bbb N$ is also discrete: if $p$ is a free (= non-principal) ultrafilter on $\Bbb N$, then $\Bbb N\in p$, so $\{p\}\cup\Bbb N$ is an open nbhd of $p$ that contains no other point of $X\setminus\Bbb N$. Thus, $X\setminus\Bbb N$ is a closed discrete subset of $X$. (And it’s a large one: there are $2^{2^\omega}=2^\mathfrak{c}$ free ultrafilters on $\Bbb N$.) However, points of $X\setminus\Bbb N$ are not isolated: $\varnothing\notin p$, so $\{p\}=\{p\}\cup\varnothing$ is not an open nbhd of $p$.

$X$ is separable, because $\Bbb N$ is a countable dense subset of $X$: every point of $X\setminus\Bbb N$ is a limit point of $\Bbb N$. However, the points of $X\setminus\Bbb N$ are limit points only of $\Bbb N$: each of them has a local base of nbhds that exclude the other points of $X\setminus\Bbb N$. Here’s a little more on the space; it’s simple stuff, but perhaps it will help you to get a better picture.

$X$ is Hausdorff.

  • If $m,n\in\Bbb N$ with $m\ne n$, then $\{m\}$ and $\{n\}$ are disjoint open nbhds of $m$ and $n$.
  • If $p\in X\setminus\Bbb N$ and $n\in\Bbb N$, then $\Bbb N\setminus\{n\}\in p$, so $\{n\}$ and $\{p\}\cup\big(\Bbb N\setminus\{n\}\big)$ are disjoint open nbhds of $n$ and $p$.
  • If $p,q\in X\setminus\Bbb N$ with $p\ne q$, then there is some $A\subseteq\Bbb N$ such that $A\in p$ and $A\notin q$. But $q$ is an ultrafilter, so $\Bbb N\setminus A\in q$. Thus, $\{p\}\cup A$ and $\{q\}\cup\big(\Bbb N\setminus A\big)$ are disjoint open nbhds of $p$ and $q$.

Each basic open set in $X$ is clopen, so $X$ is zero-dimensional. Since $X$ is also Hausdorff, this means that $X$ is Tikhonov (completely regular and $T_1$). $X$ is not normal, however; you might try to see why this is true.

Added: I should mention that the definition that you’ve been given is a little sloppy. Either $X$ should be described as the union of $\Bbb N$ and the set of free ultrafilters on $\Bbb N$, which is how I’ve treated it above, or the topology has to be defined a little differently. Specifically, if $X$ really is the set of all ultrafilters on $\Bbb N$, then the definition of the topology needs to distinguish two cases. If $p_n=\{A\subseteq\Bbb N:n\in A\}$ is the principal ultrafilter at $n\in\Bbb N$, then $\{p_n\}$ is a local base at $p_n$. Otherwise, if $p$ is free, then basic open nbhds of $p$ are the sets of the form $\{p\}\cup\{\widehat A:A\in p\}$, where $\widehat A=\{p_n:n\in A\}$.

It usually easier just to identify $p_n$ with $n$ and describe $X$ as the union of $\Bbb N$ and the set of free ultrafilters on $\Bbb N$.

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$X$ is the space of ultrafilters on $N$. How are you interpreting $N \subset X$? –  William Jun 21 '12 at 3:00
    
@William: By identification with the set of fixed ultrafilters, of course. As I indicated in my addendum, I think that the description of $X$ that John was given is very sloppy. Either the space or the topolgoy was incorrectly described. –  Brian M. Scott Jun 21 '12 at 3:06
    
@Brain thanks for your nice answer:) Is the Katetov extension of the natural numbers the same space with this space, or is another different space? –  Paul Jun 21 '12 at 3:07
    
@John: Yes, this is the Katětov extension of $\Bbb N$. –  Brian M. Scott Jun 21 '12 at 3:13
    
Thank you Brain:) –  Paul Jun 21 '12 at 3:16

The best interpretation I can come up with is :

$A$ represents the collection of principal ultra filters of elements in $A$. More precisely $[A] = \{[a] : a \in A\}$ where $[a] = \{S \subset N : a \in S\}$. Note that $[a]$ is an ultrafilter since for every $S$, either $a \in S$ or $a \in N - S$.

So I would say that the basic open sets are $\{p\} \cup [A] \subset X$, where $[A]$ is what I defined above.

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