countable well-orders

bibliography/references.bib

@@ -44,4 +44,12 @@ title = {Classical Descriptive Set Theory},
 volume = {156},
 year = {2012},
 }
+@MISC{3722713,
+    TITLE = {Embedding of countable linear orders into $\Bbb Q$ as topological spaces},
+    AUTHOR = {Eric Wofsey (https://math.stackexchange.com/users/86856/eric-wofsey)},
+    HOWPUBLISHED = {Mathematics Stack Exchange},
+    NOTE = {URL:https://math.stackexchange.com/q/3722713 (version: 2020-06-16)},
+    EPRINT = {https://math.stackexchange.com/q/3722713},
+    URL = {https://math.stackexchange.com/q/3722713}
+}
 

inputs/lecture_13.tex

@@ -18,20 +18,10 @@ by associating a function $f\colon \Q \to \{0,1\}$
 with $(f^{-1}(\{1\}), <)$.
 
 \begin{lemma}
-    Any countable ordinal embeds into $(\Q,<)$.
+    Any countable wellorder embeds into $(\Q,<)$.
 \end{lemma}
-\begin{proof}[sketch]
-    Use transfinite induction.
-    Suppose we already have $\alpha \hookrightarrow (\Q, <)$,
-    we need to show that $\alpha +1 \hookrightarrow (\Q, <)$.
-    Since $(0,1) \cap \Q \cong \Q$,
-    we may assume $\alpha \hookrightarrow ((0,1), <)$
-    and can just set $\alpha \mapsto 2$.
-
-    For a limit $\alpha$
-    take a countable cofinal subsequence $\alpha_1 < \alpha_2 < \ldots \to \alpha$.
-    Then map $[0,\alpha_1)$ to $(0,1)$
-    and  $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$.
+\begin{proof}\footnote{In the lecture this was only done for countable \emph{ordinals}.}
+    Cf.~\cite{3722713}.
 \end{proof}
 
 % TODO $\WF \subseteq 2^\Q$ is $\Sigma^1_1$-complete.

inputs/lecture_16.tex

@@ -23,7 +23,7 @@ $X$ is always compact metrizable.
     \[
         \{h^n : n \in \Z\}  \subseteq  \cC(X,X)\gist{ = \{f\colon  X \to X : f \text{ continuous}\}}{},
     \]
-    where the topology is the uniform convergence topology. % TODO REF EXERCISE
+    where the topology is the uniform convergence topology.
     Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
     Since the family $\{h^n : n \in \Z\}$ is uniformly equicontinuous,
     i.e.~