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inputs/lecture_03.tex

@@ -165,7 +165,13 @@
         Since the cantor space embeds into $X$,
         we get the lower bound.
         Since $X$ is second countable and Hausdorff,
-        we get the upper bound.%
+        we get the upper bound:
+        Let $\langle U_n : n < \omega \rangle$ be a countable basis.
+        Consider the injective function
+        \begin{IEEEeqnarray*}{rCl}
+            f\colon X &\longrightarrow & 2^{ \omega} \\
+             x &\longmapsto & \{n : x \in U_n\}.
+        \end{IEEEeqnarray*}
     }{Lower bound:  $2^{\N} \hookrightarrow X$,
         upper bound: \nth{2} countable and Hausdorff.}
 \end{proof}