From 1b05816322dc79b20976350393f71840c697eb46 Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Wed, 18 Sep 2024 15:09:51 +0200
Subject: Ambiguous parse

in urysohn2.tex line 674 the secound 2 throws an ambiguous parse.
the same definition in line 678 commented without this a does not throe this error.

ambiguous parse: [BlockLemma (SourcePos {sourceName = "set.tex", sourceLine = Pos 129, sourceColumn = Pos 1}) (Marker "empty_eq") (Lemma [] (StmtConnected Implication (StmtVerbPhrase (TermExpr (ExprVar (NamedVar "x")) :| [TermExpr (ExprVar (NamedVar "y"))]) (VPAdj (Adj [Just (Word "empty")] [] :| []))) (StmtFormula (FormulaChain (ChainBase (ExprVar (NamedVar "x") :| []) Positive (RelationSymbol (Symbol "=")) (ExprVar (NamedVar "y") :| [])))))),BlockLemma (SourcePos {sourceName = "set.tex", sourceLine = Pos 129, sourceColumn = Pos 1}) (Marker "empty_eq") (Lemma [] (StmtConnected Implication (StmtConnected Conjunction (StmtVerbPhrase (TermExpr (ExprVar (NamedVar "x")) :| []) (VPVerb (Verb (SgPl {sg = [], pl = []}) []))) (StmtVerbPhrase (TermExpr (ExprVar (NamedVar "y")) :| []) (VPAdj (Adj [Just (Word "empty")] [] :| [])))) (StmtFormula (FormulaChain (ChainBase (ExprVar (NamedVar "x") :| []) Positive (RelationSymbol (Symbol "=")) (ExprVar (NamedVar "y") :| []))))))]
---
 library/topology/urysohn2.tex | 28 +++++++++++++++++++++-------
 1 file changed, 21 insertions(+), 7 deletions(-)

(limited to 'library/topology/urysohn2.tex')

diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex
index 97bbc70..ce6d742 100644
--- a/library/topology/urysohn2.tex
+++ b/library/topology/urysohn2.tex
@@ -664,14 +664,28 @@
 \end{definition}
 
 \begin{definition}\label{staircase_sequence}
-    $f$ is staircase sequence of $U$ iff $f$ is a sequence and $U$ is a lifted urysohnchain of $X$ and $\dom{U} = \dom{f}$ and for all $n \in \dom{U}$ we have $\at{f}{n}$ is a staircase function adapted to $\at{U}{n}$ in $U$.
+    $S$ is staircase sequence of $U$ iff $S$ is a sequence and $U$ is a lifted urysohnchain of $X$ and $\dom{U} = \dom{S}$ and for all $n \in \dom{U}$ we have $\at{S}{n}$ is a staircase function adapted to $\at{U}{n}$ in $U$.
 \end{definition}
 
-\begin{definition}
-    
+\begin{definition}\label{staircase_limit_point}
+    $x$ is the staircase limit of $S$ with $y$ iff $x \in \reals$ and for all $\epsilon \in \realsplus$ there exist $n_0 \in \naturals$ such that for all $n \in \naturals$ such that $n_0 \rless n$ we have $\apply{\at{S}{n}}{y} \in \epsBall{x}{\epsilon}$.
 \end{definition}
 
+\begin{definition}\label{staircase_limit_function}
+    $f$ is a limit function of a staircase $S$ iff $S$ is staircase sequence of $U$ and $U$ is a lifted urysohnchain of $X$ and $\dom{f} = \carrier[X]$ and for all $x \in \dom{f}$ we have $f(x)$ is the staircase limit of $S$ with $x$ and $f$ is a function from $\carrier[X]$ to $\reals$.
+\end{definition}
 
+%\begin{definition}\label{staircase_limit_function}
+%    $f$ is a limit function of staircase $S$ iff $S$ is staircase sequence of $U$ and $U$ is a lifted urysohnchain of $X$ and $\dom{f} = \carrier[X]$ and for all $x \in \dom{f}$ we have $f(x)$ is the staircase limit of $S$ with $x$ and $f$ is a function from $\carrier[X]$ to $\reals$.
+%\end{definition}
+%
+%\begin{proposition}\label{staircase_limit_is_continuous}
+%    Suppose $X$ is a urysohnspace.
+%    Suppose $U$ is a lifted urysohnchain of $X$.
+%    Suppose $S$ is staircase sequence of $U$.
+%    Suppose $f$ is the limit function of a staircase $S$.
+%    Then $f$ is continuous.
+%\end{proposition}
 
 \begin{theorem}\label{urysohnsetinbeetween}
     Let $X$ be a urysohn space.
@@ -788,10 +802,10 @@
     \end{subproof}
     Take $S$ such that $S$ is staircase sequence of $U$.
 
-    For all $x \in \carrier[X]$ we have there exist $r,R$ such that $r \in \reals$ and $R$ is a sequence of reals and $\dom{R} = \naturals$ and $R$ converge to $r$ and for all $n \in \naturals$ we have $\at{R}{n} = \apply{\at{S}{n}}{x}$.
-
-    We show that for all $x \in \carrier[X]$ there exists $r \in \intervalclosed{a}{b}$ such that for .
-
+    %For all $x \in \carrier[X]$ we have there exist $r,R$ such that $r \in \reals$ and $R$ is a sequence of reals and $\dom{R} = \naturals$ and $R$ converge to $r$ and for all $n \in \naturals$ we have $\at{R}{n} = \apply{\at{S}{n}}{x}$.
+%
+    %We show that for all $x \in \carrier[X]$ there exists $r \in \intervalclosed{a}{b}$ such that for .
+%
 
     
 \end{proof}
-- 
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