diff --git a/inputs/lecture_07.tex b/inputs/lecture_07.tex
index bae467d..687e009 100644
--- a/inputs/lecture_07.tex
+++ b/inputs/lecture_07.tex
@@ -144,8 +144,9 @@
     \begin{proof}
         Consider $(F, \cT\defon{F})$ and $(X \setminus F, \cT\defon{X \setminus F})$.
         Both are Polish spaces.
-        Take the coproduct\footnote{topological sum} $F \oplus (X \setminus F)$
-        of these spaces.
+        Take the coproduct%
+        \footnote{In the lecture, this was called the \vocab{topological sum}.}
+        $F \oplus (X \setminus F)$ of these spaces.
         This space is Polish,
         and the topology is generated by $\cT \cup \{F\}$,
         hence we do not get any new Borel sets.
diff --git a/inputs/lecture_09.tex b/inputs/lecture_09.tex
index c3a61ce..ea1c2f6 100644
--- a/inputs/lecture_09.tex
+++ b/inputs/lecture_09.tex
@@ -26,7 +26,7 @@
     Let $X$ be a Polish space.
     A set $A \subseteq X$
     is called \vocab{analytic} 
-    if
+    iff
     \[
     \exists Y \text{ Polish}.~\exists  B \in \cB(Y).~
     \exists \underbrace{f\colon Y \to X}_{\text{continuous}}.~