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| -rw-r--r-- | library/topology/basis.tex | 35 |
1 files changed, 11 insertions, 24 deletions
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} - - - - - |