small changes

inputs/lecture_01.tex

@@ -42,14 +42,14 @@ Recall the following notions:
         and $U_i \subsetneq X_i$ for only finitely many $i$.
 \end{definition}
 \begin{fact}
-    Countable products of separable spaces are separable,
+    Countable products of separable spaces are separable.
 \end{fact}
 \begin{definition}
     A topological space $X$ is \vocab{second countable},
     if it has a countable base.
 \end{definition}
-If $X$ is a topological space.
-Then if $X$ is second countable, it is also separable.
+Let $X$ be a topological space.
+If $X$ is second countable, it is also separable.
 However the converse of this does not hold.
 
 \begin{example}
@@ -81,7 +81,8 @@ However the converse of this does not hold.
     For a metric space, the following are equivalent:
     \begin{itemize}
         \item compact,
-        \item \vocab{sequentially compact} (every sequence has a convergent subsequence),
+        \item \vocab{sequentially compact}
+            (every sequence has a convergent subsequence),
         \item complete and \vocab{totally bounded}
             (for all $\epsilon > 0$ we can cover the space with finitely many $\epsilon$-balls).
     \end{itemize}