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Josia Pietsch 2024-02-02 01:59:54 +01:00
parent 1cc501fbe3
commit 37f81861df
Signed by: josia
GPG key ID: E70B571D66986A2D
4 changed files with 8 additions and 10 deletions
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5
inputs/lecture_15.tex
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@ -254,15 +254,14 @@ Recall:
Hence $x_1 = x_2$ $\lightning$. Hence $x_1 = x_2$ $\lightning$.
\end{proof} \end{proof}
% TODO ANKI-MARKER
\begin{definition} \begin{definition}
Let $\Sigma = \{(X_i, T) : i \in I\} $ Let $\Sigma = \{(X_i, T) : i \in I\} $
be a collection of factors of $(X,T)$. be a collection of factors of $(X,T)$.
Let $\pi_i\colon (X,T) \to (X_i, T)$ denote the factor map. Let $\pi_i\colon (X,T) \to (X_i, T)$ denote the factor map.
Then $(X, T)$ is a \vocab{limit}% Then $(X, T)$ is a \vocab{limit}%
\footnote{This is not a limit in the category theory sense and not uniquely determined.} \footnote{This is not a limit in the category theory sense and not uniquely determined.}
of $\Sigma$ % TODO THE inverse limit is A limit
iff of $\Sigma$ iff
\[ \[
\forall x_1,x_2 \in X.~\exists i \in I.~\pi_i(x_1) \neq \pi_i(x_2). \forall x_1,x_2 \in X.~\exists i \in I.~\pi_i(x_1) \neq \pi_i(x_2).
\] \]

1
inputs/lecture_16.tex
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@ -1,4 +1,5 @@
\lecture{16}{2023-12-08}{} \lecture{16}{2023-12-08}{}
% TODO ANKI-MARKER
$X$ is always compact metrizable. $X$ is always compact metrizable.

2
inputs/tutorial_14.tex
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@ -23,7 +23,7 @@
such that for all $t \in T$, $d(ta,tb) > \epsilon$. such that for all $t \in T$, $d(ta,tb) > \epsilon$.
\item % TODO (not too hard) \item \todo{TODO}% TODO (not too hard)
% (b) % (b)
% Let $(X,T)$ be distal with a dense orbit, % Let $(X,T)$ be distal with a dense orbit,
% then it is distal minimal. % then it is distal minimal.

10
inputs/tutorial_15.tex
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@ -1,12 +1,11 @@
\tutorial{15}{2024-01-31}{Additions}
\subsection{Additional Tutorial} \subsection{Additional Tutorial}
\tutorial{15}{2024-01-31}{Additions}
The following is not relevant for the exam, The following is not relevant for the exam,
but gives a more general picture. but gives a more general picture.
Let $ X$ be a topological space. Let $X$ be a topological space
Let $\cF$ be a filter on $ X$. and let $\cF$ be a filter on $ X$.
$x \in X$ is a limit point of $\cF$ iff the neighbourhood filter $\cN_x$, $x \in X$ is a limit point of $\cF$ iff the neighbourhood filter $\cN_x$,
all sets containing an open neighbourhood of $x$, all sets containing an open neighbourhood of $x$,
@ -83,5 +82,4 @@ Consider the basic open set
\{\cF \in \beta\N : \cF \ni f^{-1}(V)\}. \{\cF \in \beta\N : \cF \ni f^{-1}(V)\}.
\] \]
% TODO