diff --git a/bibliography/references.bib b/bibliography/references.bib index 1958dc8..21b76fd 100644 --- a/bibliography/references.bib +++ b/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} +} diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index 019002f..51494d2 100644 --- a/inputs/lecture_13.tex +++ b/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. diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index 7727e51..f5ee141 100644 --- a/inputs/lecture_16.tex +++ b/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.~