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@ -93,8 +93,8 @@ where $X$ is a metrizable, usually second countable space.
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\end{proof}
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\begin{remark}
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Since $2^{\omega}$ embeds
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into any uncountable polish space $Y$
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such that the image is closed,
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into any uncountable polish space $Y$,
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% such that the image is closed,
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we can replace $2^{\omega}$ by $Y$
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in the statement of the theorem.%
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\footnote{By definition of the subspace topology
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@ -59,14 +59,14 @@
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there is a unique $x \in X$,
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such that
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\[
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(\cU_n) (x_n \in G)
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(\cU n) (x_n \in G)
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\]
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for every neighbourhood%
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\footnote{$G \subseteq X$ is a neighbourhood iff $x \in \inter G$.}
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$G$ of $x$.
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\end{lemma}
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\begin{notation}
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In this case we write $x = \cU-\lim_n x_n$.
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In this case we write $x = \ulim{\cU}_n x_n$.
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\end{notation}
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\begin{refproof}{lem:ultrafilterlimit}[sketch]
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Whenever we write $X = Y \cup Z$
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@ -101,13 +101,13 @@ This gives an operation on principal ultrafilters
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(we identify $n \in \N$ with the corresponding principal filter).
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We want to extend this to all of $\beta\N$.
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Fix the first argument to get a function $\N \to \N, n \mapsto k+n$.
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For $\cU \in \beta\N$ consider $\cU-\lim_n (k+n)$.
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For $\cU \in \beta\N$ consider $\ulim{\cU}_n (k+n)$.
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So for a fixed $k \in \N$ we get $k+ \cdot \colon\beta\N \to \beta\N$,
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i.e.~$+ \colon \N \times \beta\N \to \beta\N$.
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Fixing the second coordinate to be $\cV \in \beta\N$,
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we get a function $+\cV \colon \N \to \beta\N$.
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For $ \cU \in \beta\N$
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consider $\cU-\lim_n n + \cV$.
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consider $\ulim{\cU}_n n + \cV$.
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This gives $+ \colon \beta\N \times \beta\N \to \beta\N$.
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% TODO ?
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@ -130,7 +130,7 @@ and let $T \colon X \to X$ be continuous.%
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but not a $\Z$-action.}
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For any $\cU \in \beta\N$, we define $T^{\cU}$ by
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$T^\cU(x) \coloneqq \cU-\lim_n T^n(x)$ for $x \in X$.
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$T^\cU(x) \coloneqq \ulim{\cU}_n T^n(x)$ for $x \in X$.
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For fixed $x$, the map $\cU \mapsto T^{\cU}(x)$ is continuous.
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@ -165,9 +165,9 @@ is not necessarily continuous.
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\end{fact}
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\begin{proof}
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\begin{IEEEeqnarray*}{rCl}
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T^{\cU + \cV}(x) &=& (\cU + \cV)-\lim_k T^k(x)\\
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&=& \cU-\lim_m \cV-\lim_n T^{m+n}(x)\\
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&\overset{T^m \text{ continuous}}{=}& \cU-\lim_m T^m (\cV-\lim_n T^n(x))\\
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T^{\cU + \cV}(x) &=& \ulim{(\cU + \cV)}_k T^k(x)\\
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&=& \ulim{\cU}_m \ulim{\cV}_n T^{m+n}(x)\\
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&\overset{T^m \text{ continuous}}{=}& \ulim{\cU}_m T^m (\ulim{\cV}_n T^n(x))\\
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&=& T^\cU(T^\cV(x)).
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\end{IEEEeqnarray*}
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\end{proof}
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@ -86,7 +86,7 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$.
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For every compact Hausdorff space $X$,
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a sequence $(x_n)$ in $X$,
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and $\cU \in \beta\N$,
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we have that $\cU-\lim_n x_n = x$
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we have that $\ulim{\cU}_n x_n = x$
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exists and is unique,
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i.e.~for all $x \in G \overset{\text{open}}{\subseteq} X$
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we have $\{n \in \N : x_n \in G\} \in \cU$.
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@ -110,7 +110,7 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$.
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\end{IEEEeqnarray*}
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since $B_1 \cup \ldots \cup B_m \in \cU \iff \exists i < m.~B_i \in \cU$.
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It is clear that $\cU-\lim_n x_n$
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It is clear that $\ulim{\cU}_n x_n$
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is unique, since $X$ is Hausdorff.
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\end{proof}
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@ -123,7 +123,7 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$.
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Let
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\begin{IEEEeqnarray*}{rCl}
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\tilde{f}\colon \beta\N &\longrightarrow & X \\
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\cU &\longmapsto & \cU-\lim_n f(n).
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\cU &\longmapsto & \ulim{\cU}_n f(n).
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\end{IEEEeqnarray*}
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\todo{Exercise: Check that $\tilde{f}$ is continuous.}
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@ -189,7 +189,7 @@ we need the following:
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(\cV n) T^n(x) \in \underbrace{T^{-k}(\overbrace{G}^{\text{closed}})}_{\text{closed}}.
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\]
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Therefore $\cV-\lim_n T^n(x) \in T^{-k}(G)$.
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Therefore $\ulim{\cV}_n T^n(x) \in T^{-k}(G)$.
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So $T^k(T^\cV(x)) \in G \subseteq G_0$.
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@ -30,7 +30,7 @@
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\]
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Thus
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\[
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\underbrace{\cV-\lim_n T^n(x)}_{T^\cV(x)} \not\in T^{-k}(G).
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\underbrace{\ulim{\cV}_n T^n(x)}_{T^\cV(x)} \not\in T^{-k}(G).
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\]
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We get that
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\[
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@ -48,8 +48,8 @@ $S \colon \beta\N \to \beta\N$,
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$S(\cU ) = \hat{1}+ \cU$.
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Then
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\[
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S^\cV(\cU) = \cV-\lim_n S^n(\cU) = \cV-\lim_n(\hat{n} + \cU) =
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\cV-\lim_n \hat{n} + \cU = \cV + \cU.
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S^\cV(\cU) = \ulim{\cV}_n S^n(\cU) = \ulim{\cV}_n(\hat{n} + \cU) =
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\ulim{\cV}_n \hat{n} + \cU = \cV + \cU.
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\]
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% TODO check
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@ -75,8 +75,6 @@ S^\cV(\cU) = \cV-\lim_n S^n(\cU) = \cV-\lim_n(\hat{n} + \cU) =
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\[
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\forall \cU \in I.~\forall \cV \in \beta\N.~\cV + \cU \in I.
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\]
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\end{definition}
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\begin{theorem}
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@ -151,6 +151,7 @@
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\DeclareSimpleMathOperator{proj}
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\newcommand{\fc}{\mathfrak{c}}
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\DeclareMathOperator{\acts}{\curvearrowright}
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\newcommand{\ulim}[1]{\mathop{#1\text{-lim}}\limits}
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\newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}
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\newcommand\tutorial[3]{\hrule{\color{darkgray}\hfill{\tiny[Tutorial #1, #2]}}}
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