ultrafilter limit

inputs/lecture_08.tex

@@ -93,8 +93,8 @@ where $X$ is a metrizable, usually second countable space.
 \end{proof}
 \begin{remark}
     Since $2^{\omega}$ embeds
-    into any uncountable polish space $Y$
-    such that the image is closed,
+    into any uncountable polish space $Y$,
+    % such that the image is closed,
     we can replace $2^{\omega}$ by $Y$
     in the statement of the theorem.%
     \footnote{By definition of the subspace topology

inputs/lecture_24.tex

@@ -59,14 +59,14 @@
     there is a unique $x \in X$,
     such that
     \[
-    (\cU_n) (x_n \in G)
+    (\cU n) (x_n \in G)
     \]
     for every neighbourhood%
     \footnote{$G \subseteq X$ is a neighbourhood iff $x \in \inter G$.}
     $G$ of $x$.
 \end{lemma}
 \begin{notation}
-    In this case we write $x = \cU-\lim_n x_n$.
+    In this case we write $x = \ulim{\cU}_n x_n$.
 \end{notation}
 \begin{refproof}{lem:ultrafilterlimit}[sketch]
     Whenever we write $X = Y \cup Z$
@@ -101,13 +101,13 @@ This gives an operation on principal ultrafilters
 (we identify $n \in \N$ with the corresponding principal filter).
 We want to extend this to all of $\beta\N$.
 Fix the first argument to get a function $\N \to  \N, n \mapsto k+n$.
-For $\cU \in \beta\N$ consider $\cU-\lim_n (k+n)$.
+For $\cU \in \beta\N$ consider $\ulim{\cU}_n (k+n)$.
 So for a fixed $k \in \N$ we get $k+ \cdot \colon\beta\N \to  \beta\N$,
 i.e.~$+ \colon \N \times  \beta\N \to  \beta\N$.
 Fixing the second coordinate to be $\cV \in \beta\N$,
 we get a function $+\cV \colon \N \to \beta\N$.
 For $ \cU \in \beta\N$
-consider $\cU-\lim_n n + \cV$.
+consider $\ulim{\cU}_n n + \cV$.
 This gives $+ \colon \beta\N \times  \beta\N \to  \beta\N$.
 % TODO ?
 
@@ -130,7 +130,7 @@ and let $T \colon  X \to  X$ be continuous.%
     but not a $\Z$-action.}
 
 For any $\cU \in \beta\N$, we define $T^{\cU}$ by
-$T^\cU(x) \coloneqq  \cU-\lim_n T^n(x)$ for $x \in X$.
+$T^\cU(x) \coloneqq  \ulim{\cU}_n T^n(x)$ for $x \in X$.
 
 For fixed $x$, the map $\cU \mapsto T^{\cU}(x)$ is continuous.
 
@@ -165,9 +165,9 @@ is not necessarily continuous.
 \end{fact}
 \begin{proof}
     \begin{IEEEeqnarray*}{rCl}
-        T^{\cU + \cV}(x) &=& (\cU + \cV)-\lim_k T^k(x)\\
-                         &=& \cU-\lim_m \cV-\lim_n T^{m+n}(x)\\
-                         &\overset{T^m \text{ continuous}}{=}& \cU-\lim_m T^m (\cV-\lim_n T^n(x))\\
+        T^{\cU + \cV}(x) &=& \ulim{(\cU + \cV)}_k T^k(x)\\
+                         &=& \ulim{\cU}_m \ulim{\cV}_n T^{m+n}(x)\\
+                         &\overset{T^m \text{ continuous}}{=}& \ulim{\cU}_m T^m (\ulim{\cV}_n T^n(x))\\
                          &=& T^\cU(T^\cV(x)).
     \end{IEEEeqnarray*}
 \end{proof}

inputs/lecture_25.tex

@@ -86,7 +86,7 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$.
     For every compact Hausdorff space $X$,
     a sequence $(x_n)$ in $X$,
     and $\cU \in \beta\N$,
-    we have that $\cU-\lim_n x_n = x$
+    we have that $\ulim{\cU}_n x_n = x$
     exists and is unique,
     i.e.~for all  $x \in G \overset{\text{open}}{\subseteq} X$
     we have $\{n \in \N : x_n \in G\}  \in \cU$.
@@ -110,7 +110,7 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$.
     \end{IEEEeqnarray*}
     since $B_1 \cup \ldots \cup B_m \in \cU \iff \exists  i < m.~B_i \in \cU$.
 
-    It is clear that  $\cU-\lim_n x_n$
+    It is clear that  $\ulim{\cU}_n x_n$
     is unique, since  $X$ is Hausdorff.
 \end{proof}
 
@@ -123,7 +123,7 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$.
     Let
     \begin{IEEEeqnarray*}{rCl}
         \tilde{f}\colon \beta\N &\longrightarrow & X \\
-        \cU &\longmapsto & \cU-\lim_n f(n).
+        \cU &\longmapsto & \ulim{\cU}_n f(n).
     \end{IEEEeqnarray*}
     \todo{Exercise: Check that $\tilde{f}$ is continuous.}
 

inputs/lecture_26.tex

@@ -189,7 +189,7 @@ we need the following:
             (\cV n) T^n(x) \in  \underbrace{T^{-k}(\overbrace{G}^{\text{closed}})}_{\text{closed}}.
         \]
 
-        Therefore $\cV-\lim_n T^n(x) \in T^{-k}(G)$.
+        Therefore $\ulim{\cV}_n T^n(x) \in T^{-k}(G)$.
         So $T^k(T^\cV(x)) \in G \subseteq G_0$.
 
 

inputs/lecture_27.tex

@@ -30,7 +30,7 @@
         \]
         Thus
         \[
-            \underbrace{\cV-\lim_n T^n(x)}_{T^\cV(x)} \not\in T^{-k}(G).
+            \underbrace{\ulim{\cV}_n T^n(x)}_{T^\cV(x)} \not\in T^{-k}(G).
         \]
         We get that
         \[
@@ -48,8 +48,8 @@ $S \colon \beta\N \to \beta\N$,
 $S(\cU ) = \hat{1}+ \cU$.
 Then
 \[
-S^\cV(\cU) = \cV-\lim_n S^n(\cU) = \cV-\lim_n(\hat{n} + \cU) =
-\cV-\lim_n \hat{n} + \cU = \cV + \cU.
+S^\cV(\cU) = \ulim{\cV}_n S^n(\cU) = \ulim{\cV}_n(\hat{n} + \cU) =
+\ulim{\cV}_n \hat{n} + \cU = \cV + \cU.
 \]
 % TODO check
 
@@ -75,8 +75,6 @@ S^\cV(\cU) = \cV-\lim_n S^n(\cU) = \cV-\lim_n(\hat{n} + \cU) =
     \[
     \forall  \cU \in I.~\forall  \cV \in \beta\N.~\cV + \cU \in I.
     \]
-    
-    
 \end{definition}
 
 \begin{theorem}

logic.sty

@@ -151,6 +151,7 @@
 \DeclareSimpleMathOperator{proj}
 \newcommand{\fc}{\mathfrak{c}}
 \DeclareMathOperator{\acts}{\curvearrowright}
+\newcommand{\ulim}[1]{\mathop{#1\text{-lim}}\limits}
 
 \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}
 \newcommand\tutorial[3]{\hrule{\color{darkgray}\hfill{\tiny[Tutorial #1, #2]}}}