diff --git a/inputs/lecture_07.tex b/inputs/lecture_07.tex index 169fc76..bae467d 100644 --- a/inputs/lecture_07.tex +++ b/inputs/lecture_07.tex @@ -77,11 +77,18 @@ \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} Let $(X, \cT)$ be a Polish space. 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} such that \begin{itemize}