From 3ae13f68d8d572aa62982bde614acf79dff60848 Mon Sep 17 00:00:00 2001
From: adelon <22380201+adelon@users.noreply.github.com>
Date: Tue, 28 May 2024 17:27:29 +0200
Subject: Update `inters_in_genopens`

---
 library/topology/basis.tex | 35 +++++++++++------------------------
 1 file changed, 11 insertions(+), 24 deletions(-)

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

diff --git a/library/topology/basis.tex b/library/topology/basis.tex
index 15910f9..4f66166 100644
--- a/library/topology/basis.tex
+++ b/library/topology/basis.tex
@@ -75,45 +75,32 @@
 \end{proof}
 
 
-
-
 \begin{lemma}\label{inters_in_genopens}
     Assume $B$ is a topological basis for $X$.
-    %For all $A, C$ 
-    If $A\in \genOpens{B}{X}$ and $C\in \genOpens{B}{X}$ then $(A\inter C) \in \genOpens{B}{X}$.
+    Assume $A, C\in \genOpens{B}{X}$.
+    Then $(A\inter C) \in \genOpens{B}{X}$.
 \end{lemma}
 \begin{proof}
-    
-    Show $(A \inter C) \in \pow{X}$.
-    \begin{subproof}
-        $(A \inter C) \subseteq X$ by assumption.
-    \end{subproof}
-    Therefore for all $A, C \in \genOpens{B}{X}$ we have $(A \inter C) \in \pow{X}$.
-    
+
+    We have $(A \inter C) \in \pow{X}$ by \cref{genopens,inter_powerset}.
 
     Show for all $x\in (A\inter C)$ there exists $W \in B$
     such that $x\in W$ and $W \subseteq (A\inter C)$.
     \begin{subproof}
         Fix $x \in (A\inter C)$.
-        There exist $V' \in B$ such that $x \in V'$ and $V' \subseteq A$ by assumption.  %TODO: Warum muss hier by assumtion hin?
-        There exist $V'' \in B$ such that $x \in V''$ and $V'' \subseteq C$ by assumption.
-        There exist $W \in B$ such that $x \in W$ and $W \subseteq v'$ and $W \subseteq V''$ by assumption.
-        
+        Then $x\in A,C$.
+        There exists $V'  \in B$ such that $x \in V' \subseteq A$ by \cref{genopens}.
+        There exists $V'' \in B$ such that $x \in V''\subseteq C$ by \cref{genopens}.
+        There exists $W \in B$ such that $x \in W$ and $W \subseteq V'$ and $W \subseteq V''$.
+
         Show $W \subseteq (A\inter C)$.
         \begin{subproof}
             %$W \subseteq v'$ and $W \subseteq V''$.
-            For all $y \in W$ we have $y \in V'$ and $y \in V''$ by assumption.
+            For all $y \in W$ we have $y \in V'$ and $y \in V''$.
         \end{subproof}
     \end{subproof}
     %Therefore for all $A, C, x$ such that $A \in \genOpens{B}{X}$ and $C \in \genOpens{B}{X}$ and $x \in (A \inter C)$ we have there exists $W \in B$
     %such that $x\in W$ and $W \subseteq (A\inter C)$.
 
-    $(A\inter C) \in \genOpens{B}{X}$ by assumption.
-    
- 
+    $(A\inter C) \in \genOpens{B}{X}$.
 \end{proof}
-
-
-
-
-
-- 
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