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clopenization
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Josia Pietsch 2024-01-12 01:31:12 +01:00
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inputs/lecture_07.tex
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@ -77,11 +77,18 @@
\subsection{Turning Borels Sets into Clopens} \subsection{Turning Borels Sets into Clopens}
\begin{theorem} \begin{theorem}%
\footnote{Whilst strikingly concise the verb ``\vocab[Clopenization™]{to clopenize}''
unfortunately seems to be non-standard vocabulary.
Our tutor repeatedly advised against using it in the final exam.
Contrary to popular belief
the very same tutor was \textit{not} the one first to introduce it,
as it would certainly be spelled ``to clopenise'' if that were the case.
}
\label{thm:clopenize} \label{thm:clopenize}
Let $(X, \cT)$ be a Polish space. Let $(X, \cT)$ be a Polish space.
For any Borel set $A \subseteq X$, For any Borel set $A \subseteq X$,
there is a finer Polish topology, there is a finer Polish topology,%
\footnote{i.e.~$\cT_A \supseteq \cT$ and $(X, \cT_A)$ is Polish} \footnote{i.e.~$\cT_A \supseteq \cT$ and $(X, \cT_A)$ is Polish}
such that such that
\begin{itemize} \begin{itemize}