1. ## Ultrafilters Topology

Let $\displaystyle X$ be a topological space. let $\displaystyle X'$ be the collection of all the ultrafilters on $\displaystyle X$.
we say that a set $\displaystyle S\subseteq X'$ is open and belongs to the base of $\displaystyle X'$ if

there is an open $\displaystyle U\subseteq X$, such that $\displaystyle U'\in S$ $\displaystyle \Leftrightarrow$ $\displaystyle U\in U'$.

Let $\displaystyle N'$ be the space of ultrafilters (filters on $\displaystyle N$ with the discrete topology) as explained above.
Prove that if a sequence $\displaystyle U_{i \ i\in N}$ converge in $\displaystyle N' \ \ U_i=U_j$ for every large enough $\displaystyle i,j$.

any help is appreciated...

2. Originally Posted by aharonidan
Let $\displaystyle X$ be a topological space. let $\displaystyle X'$ be the collection of all the ultrafilters on $\displaystyle X$.
we say that a set $\displaystyle S\subseteq X'$ is open and belongs to the base of $\displaystyle X'$ if

there is an open $\displaystyle U\subseteq X$, such that $\displaystyle U'\in S$ $\displaystyle \Leftrightarrow$ $\displaystyle U\in U'$.

Let $\displaystyle N'$ be the space of ultrafilters (filters on $\displaystyle N$ with the discrete topology) as explained above.
Prove that if a sequence $\displaystyle U_{i \ i\in N}$ converge in $\displaystyle N' \ \ U_i=U_j$ for every large enough $\displaystyle i,j$.

any help is appreciated...
Maybe I'm being deceived, but isn't this just a consequence that for any discrete space $\displaystyle X$ the only convergent sequences are eventually constant?

3. Originally Posted by aharonidan
Let $\displaystyle X$ be a topological space. let $\displaystyle X'$ be the collection of all the ultrafilters on $\displaystyle X$.
we say that a set $\displaystyle S\subseteq X'$ is open and belongs to the base of $\displaystyle X'$ if

there is an open $\displaystyle U\subseteq X$, such that $\displaystyle U'\in S$ $\displaystyle \Leftrightarrow$ $\displaystyle U\in U'$.

Let $\displaystyle N'$ be the space of ultrafilters (filters on $\displaystyle N$ with the discrete topology) as explained above.
Prove that if a sequence $\displaystyle U_{i \ i\in N}$ converge in $\displaystyle N' \ \ U_i=U_j$ for every large enough $\displaystyle i,j$.

any help is appreciated...
Another way to phrase this problem is to say that every convergent sequence in the Stone–Čech compactification of N is eventually constant. I believe that this is a standard result in the theory of the Stone–Čech compactification, but I do not know of a proof. You might try looking here for ideas and further references.