From ecfb1a66f2159e078199e54edf8a80004c28195a Mon Sep 17 00:00:00 2001 From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com> Date: Tue, 28 May 2024 15:33:45 +0200 Subject: proofing some lammes about topological basis --- library/topology/basis.tex | 75 +++++++++++++++++++++++++++++++++++++-- library/topology/metric-space.tex | 60 ++++++++++++++++++++----------- 2 files changed, 111 insertions(+), 24 deletions(-) (limited to 'library/topology') diff --git a/library/topology/basis.tex b/library/topology/basis.tex index bca42f0..61a358f 100644 --- a/library/topology/basis.tex +++ b/library/topology/basis.tex @@ -1,4 +1,6 @@ \import{topology/topological-space.tex} +\import{set.tex} +\import{set/powerset.tex} \subsection{Topological basis} @@ -44,7 +46,74 @@ there exists $W\in B$ such that $x\in W\subseteq U, V$. \end{definition} -\begin{definition}\label{genOpens} - $\genOpens{B}{X} = \{ U\in\pow{X} \mid for all $x\in U$ there exists $V\in B$ - such that $x\in V\subseteq U$\}$. +\begin{definition}\label{genopens} + $\genOpens{B}{X} = \{ U\in\pow{X} \mid \text{for all $x\in U$ there exists $V\in B$ + such that $x\in V\subseteq U$} \}$. \end{definition} + +\begin{lemma}\label{emptyset_in_genopens} + Assume $B$ is a topological basis for $X$. + $\emptyset \in \genOpens{B}{X}$. +\end{lemma} + +\begin{lemma}\label{all_is_in_genopens} + Assume $B$ is a topological basis for $X$. + $X \in \genOpens{B}{X}$. +\end{lemma} +\begin{proof} + $B$ covers $X$ by \cref{topological_prebasis_iff_covering_family,topological_basis}. + $\unions{B} \in \genOpens{B}{X}$. + $X \subseteq \unions{B}$. +\end{proof} + +\begin{lemma}\label{union_in_genopens} + Assume $B$ is a topological basis for $X$. + For all $F\subseteq \genOpens{B}{X}$ we have $\unions{F}\in\genOpens{B}{X}$. +\end{lemma} +\begin{proof} + Omitted. +\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 $(A\inter C) \in \genOpens{B}{X}$. +\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 $(A \inter C) \in \pow{X}$. + + + 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. + + 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. + \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. + + +\end{proof} + + + + + diff --git a/library/topology/metric-space.tex b/library/topology/metric-space.tex index 8ec83f7..1c6a0ca 100644 --- a/library/topology/metric-space.tex +++ b/library/topology/metric-space.tex @@ -2,6 +2,7 @@ \import{numbers.tex} \import{function.tex} \import{set/powerset.tex} +\import{topology/basis.tex} \section{Metric Spaces} @@ -20,38 +21,42 @@ -\begin{definition}\label{induced_topology} - $O$ is the induced topology of $d$ in $M$ iff - $O \subseteq \pow{M}$ and - $d$ is a metric on $M$ and - for all $x,r,A,B,C$ - such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$ - we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$. -\end{definition} +%\begin{definition}\label{induced_topology} +% $O$ is the induced topology of $d$ in $M$ iff +% $O \subseteq \pow{M}$ and +% $d$ is a metric on $M$ and +% for all $x,r,A,B,C$ +% such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$ +% we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$. +%\end{definition} %\begin{definition} % $\projcetfirst{A} = \{a \mid \exists x \in X \text{there exist $x \i } \}$ %\end{definition} \begin{definition}\label{set_of_balls} - $\balls{d}{M} = \{ O \in \pow{M} \mid \text{there exists $x,r$ such that $r \in \reals$ and $x \in M$ we have $O = \openball{r}{x}{d}$ } \}$. + $\balls{d}{M} = \{ O \in \pow{M} \mid \text{there exists $x,r$ such that $r \in \reals$ and $x \in M$ and $O = \openball{r}{x}{d}$ } \}$. \end{definition} -\begin{definition}\label{toindsas} - $\metricopens{d}{M} = \{O \in \pow{M} \mid \text{ - $d$ is a metric on $M$ and - for all $x,r,A,B,C$ - such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$ - we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$. - } \}$. - -\end{definition} +%\begin{definition}\label{toindsas} +% $\metricopens{d}{M} = \{O \in \pow{M} \mid \text{ +% $d$ is a metric on $M$ and +% for all $x,r,A,B,C$ +% such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$ +% we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$. +% } \}$. +% +%\end{definition} -\begin{theorem}\label{metric_induce_a_topology} - +\begin{definition}\label{metricopens} + $\metricopens{d}{M} = \genOpens{\balls{d}{M}}{M}$. +\end{definition} +\begin{theorem} + Let $d$ be a metric on $M$. + $M$ is a topological space. \end{theorem} @@ -70,7 +75,7 @@ \begin{enumerate} \item \label{metric_space_metric} $\metric[M]$ is a metric on $M$. \item \label{metric_space_topology} $M$ is a topological space. - \item \label{metric_space_opens} for all $x \in M$ for all $r \in \reals$ $\openball{r}{x}{\metric[M]} \in \opens[M]$. + \item \label{metric_space_opens} $\metricopens{ \metric[M] }{M} = \opens[M]$. \end{enumerate} \end{struct} @@ -132,3 +137,16 @@ % Then $\openball{r}{x}{M} \in \opens[M]$. %\end{proposition} + + + + + +%TODO: - Basis indudiert topology lemma +% - Offe Bälle sind basis + +% Was danach kommen soll bleibt offen, vll buch oder in proof wiki +% Trennungsaxiom, + +% Notaionen aufräumen damit das gut gemercht werden kann. + -- cgit v1.2.3