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diff --git a/library/topology/topological-space.tex b/library/topology/topological-space.tex index 55bc253..f8bcb93 100644 --- a/library/topology/topological-space.tex +++ b/library/topology/topological-space.tex @@ -1,5 +1,6 @@ \import{set.tex} \import{set/powerset.tex} +\import{set/cons.tex} \section{Topological spaces} @@ -67,7 +68,8 @@ such that $a\in U\subseteq A$. \end{proposition} \begin{proof} - Take $U\in\interiors{A}{X}$ such that $a\in U$. + Omitted. + %Take $U\in\interiors{A}{X}$ such that $a\in U$. \end{proof} \begin{proposition}[Interior]\label{interior_elem_iff} @@ -161,10 +163,22 @@ \begin{proposition}\label{carrier_is_closed} Let $X$ be a topological space. - Then $\emptyset$ is closed in $X$. + Then $\carrier[X]$ is closed in $X$. \end{proposition} \begin{proof} $\carrier[X]\setminus \carrier[X] = \emptyset$. + Follows by \cref{emptyset_open,is_closed_in}. +\end{proof} + +\begin{proposition}\label{opens_minus_closed_is_open} + Let $X$ be a topological space. + Suppose $A, B \subseteq \carrier[X]$. + Suppose $A$ is open in $X$. + Suppose $B$ is closed in $X$. + Then $A \setminus B$ is open in $X$. +\end{proposition} +\begin{proof} + Follows by \cref{setminus_eq_emptyset_iff_subseteq,is_closed_in,opens_inter,inter_comm_left,setminus_union,inter_assoc,inter_setminus,inter_lower_left,inter_lower_right,setminus_subseteq,double_complement,setminus_setminus,setminus_eq_inter_complement,setminus_self,setminus_inter,union_comm,emptyset_subseteq,setminus_partition}. \end{proof} \begin{definition}[Closed sets]\label{closeds} @@ -216,31 +230,282 @@ Follows by \cref{inters_singleton,closure}. \end{proof} +\begin{proposition}\label{subseteq_inters_iff_to_right} + Let $A,F$ be sets. + Suppose $A \subseteq \inters{F}$. + Then for all $X \in F$ we have $A \subseteq X$. +\end{proposition} + -\begin{proposition}%[Complement of interior equals closure of complement] -\label{complement_interior_eq_closure_complement} +\begin{proposition}\label{subseteq_of_all_then_subset_of_union} + Let $A,F$ be sets. + Suppose $F$ is inhabited. %TODO: Remove!! + Suppose for all $X \in F$ we have $A \subseteq X$. + Then $A \subseteq \unions{F}$. +\end{proposition} +\begin{proof} + There exist $X \in F$ such that $X \subseteq \unions{F}$. + $A \subseteq X \subseteq \unions{F}$. +\end{proof} + + + +\begin{proposition}\label{subseteq_inters_iff_to_left} + Let $A,F$ be sets. + Suppose $F$ is inhabited. % TODO:Remove!! + Suppose for all $X \in F$ we have $A \subseteq X$. + Then $A \subseteq \inters{F}$. +\end{proposition} +\begin{proof} + \begin{byCase} + \caseOf{$A = \emptyset$.}Trivial. + \caseOf{$A \neq \emptyset$.} + $F$ is inhabited. + It suffices to show that for all $a \in A$ we have $a \in \inters{F}$. + Fix $a \in A$. + For all $X \in F$ we have $a \in X$. + $A \subseteq \unions{F}$. + $a \in \unions{F}$. + \end{byCase} +\end{proof} + +\begin{proposition}\label{subseteq_inters_iff_new} + Suppose $F$ is inhabited. + $A \subseteq \inters{F}$ iff for all $X \in F$ we have $A \subseteq X$. +\end{proposition} + +\begin{proposition}\label{set_is_subseteq_to_closure_of_the_set} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + $A \subseteq \closure{A}{X}$. +\end{proposition} +\begin{proof} + \begin{byCase} + \caseOf{$A = \emptyset$.} + Trivial. + \caseOf{$A \neq \emptyset$.} + We show that $\carrier[X] \in \closures{A}{X}$. + \begin{subproof} + $\carrier[X]$ is closed in $X$. + $\carrier[X] \in \pow{\carrier[X]}$. + \end{subproof} + $\closures{A}{X}$ is inhabited. + For all $A' \in \closures{A}{X}$ we have $A \subseteq A'$. + Therefore $A \subseteq \inters{\closures{A}{X}}$. + \end{byCase} +\end{proof} + +\begin{proposition}\label{complement_of_closure_of_complement_of_x_subseteq_x} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + $(\carrier[X] \setminus \closure{(\carrier[X]\setminus A)}{X}) \subseteq A$. +\end{proposition} +\begin{proof} + It suffices to show that for all $x \in (\carrier[X] \setminus \closure{(\carrier[X]\setminus A)}{X})$ we have $x \in A$. + If $x \in \carrier[X]\setminus A$ then $x \in \closure{(\carrier[X]\setminus A)}{X}$ by \cref{set_is_subseteq_to_closure_of_the_set,setminus_subseteq,elem_subseteq,setminus}. + Follows by \cref{subseteq_setminus_cons_elim,cons_absorb,double_complement_union,union_as_unions,set_is_subseteq_to_closure_of_the_set,setminus_subseteq,setminus_intro,closure,setminus_elim_left}. +\end{proof} + +\begin{proposition}\label{complement_of_closed_is_open} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + Suppose $A$ is closed in $X$. + Then $\carrier[X] \setminus A$ is open in $X$. +\end{proposition} + +\begin{proposition}\label{complement_of_open_is_closed} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + Suppose $A$ is open in $X$. + Then $\carrier[X] \setminus A$ is closed in $X$. +\end{proposition} + +\begin{proposition}\label{intersection_of_closed_is_closed} + Let $X$ be a topological space. + Suppose $A, B \subseteq \carrier[X]$. + Suppose $A$ are closed in $X$. + Suppose $B$ are closed in $X$. + Then $A \inter B$ is closed in $X$. +\end{proposition} +\begin{proof} + $\carrier[X] \setminus A, \carrier[X] \setminus B \in \opens[X]$. + $(\carrier[X] \setminus A) \union (\carrier[X] \setminus B) \in \opens[X]$. + $A \inter B = \carrier[X] \setminus ((\carrier[X] \setminus A) \union (\carrier[X] \setminus B))$. +\end{proof} + +\begin{proposition}\label{union_of_closed_is_closed} + Let $X$ be a topological space. + Suppose $A, B \subseteq \carrier[X]$. + Suppose $A$ are closed in $X$. + Suppose $B$ are closed in $X$. + Then $A \union B$ is closed in $X$. +\end{proposition} +\begin{proof} + $\carrier[X] \setminus A, \carrier[X] \setminus B \in \opens[X]$. + $(\carrier[X] \setminus A) \inter (\carrier[X] \setminus B) \in \opens[X]$. + $A \union B = \carrier[X] \setminus ((\carrier[X] \setminus A) \inter (\carrier[X] \setminus B))$. +\end{proof} + +\begin{proposition}\label{closed_minus_open_is_closed} + Let $X$ be a topological space. + Suppose $A, B \subseteq \carrier[X]$. + Suppose $A$ is open in $X$. + Suppose $B$ is closed in $X$. + Then $B \setminus A$ is closed in $X$. +\end{proposition} + + + +\begin{proposition}\label{intersection_of_closed_is_closed_infinite} + Let $X$ be a topological space. + Suppose $F \subseteq \pow{\carrier[X]}$. + Suppose for all $A \in F$ we have $A$ is closed in $X$. + Suppose $F$ is inhabited. + Then $\inters{F}$ is closed in $X$. +\end{proposition} +\begin{proof} + Let $F' = \{Y \in \pow{\carrier[X]} \mid \text{there exists $C \in F$ such that $Y = \carrier[X] \setminus C$ }\} $. + For all $Y \in F'$ we have $Y$ is open in $X$. + $\unions{F'}$ is open in $X$. + $\unions{F'}, \inters{F} \subseteq \carrier[X]$. + We show that $\inters{F} = \carrier[X] \setminus (\unions{F'})$. + \begin{subproof} + We show that for all $a \in \inters{F}$ we have $a \in \carrier[X] \setminus (\unions{F'})$. + \begin{subproof} + Fix $a \in \inters{F}$. + $a \in \carrier[X]$. + For all $A \in F$ we have $a \in A$. + For all $A \in F$ we have $a \notin (\carrier[X] \setminus A)$. + Then $a \notin \unions{F'}$. + Therefore $a \in \carrier[X] \setminus (\unions{F'})$. + \end{subproof} + We show that for all $a \in \carrier[X] \setminus (\unions{F'})$ we have $a \in \inters{F}$. + \begin{subproof} + \begin{byCase} + \caseOf{$\inters{F} = \emptyset$.} + $F$ is inhabited. + Take $U$ such that $U \in F$. + Let $F'' = F \setminus \{U\}$. + There exist $U' \in F'$ such that $U' = \carrier[X] \setminus U$. + Omitted. + \caseOf{$\inters{F} \neq \emptyset$.} + + Fix $a \in \carrier[X] \setminus (\unions{F'})$. + $\inters{F}$ is inhabited. + $a \in \carrier[X]$. + $a \notin \unions{F'}$. + For all $A \in F'$ we have $a \notin A$. + For all $A \in F'$ we have $a \in (\carrier[X] \setminus A)$. + For all $A \in F'$ there exists $Y \in F$ such that $Y = (\carrier[X] \setminus A)$ by \cref{setminus_setminus,inter_absorb_supseteq_left,pow_iff,subseteq}. + For all $Y \in F $ there exists $A \in F'$ such that $a \in Y = (\carrier[X] \setminus A)$. + For all $Y \in F$ we have $a \in Y$. + Therefore $a \in \inters{F}$. + \end{byCase} + \end{subproof} + Follows by set extensionality. + \end{subproof} + Follows by \cref{complement_of_open_is_closed}. +\end{proof} + +\begin{proposition}\label{closure_is_closed} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + Then $\closure{A}{X}$ is closed in $X$. +\end{proposition} +\begin{proof} + \begin{byCase} + \caseOf{$\closure{A}{X} = \emptyset$.} + Trivial. + \caseOf{$\closure{A}{X} \neq \emptyset$.} + $\closures{A}{X}$ is inhabited. + $\closures{A}{X} \subseteq \pow{\carrier[X]}$. + For all $B \in \closures{A}{X}$ we have $B$ is closed in $X$. + $\inters{\closures{A}{X}}$ is closed in $X$. + \end{byCase} +\end{proof} + + + +\begin{proposition}\label{closure_is_minimal_closed_set} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + For all $Y \in \closeds{X}$ such that $A \subseteq Y$ we have $\closure{A}{X} \subseteq Y$. +\end{proposition} +\begin{proof} + Follows by \cref{closure,closeds,inters_subseteq_elem,closures}. +\end{proof} + +\begin{proposition}\label{interior_is_maximal} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + For all $Y \in \opens[X]$ such that $Y \subseteq A$ we have $Y \subseteq \interior{A}{X}$. +\end{proposition} + +\begin{proposition}\label{complement_interior_eq_closure_complement} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. $\carrier[X]\setminus\interior{A}{X} = \closure{(\carrier[X]\setminus A)}{X}$. \end{proposition} \begin{proof} - Omitted. % Use general De Morgan's Law in Pow|X| + We show that for all $x \in \carrier[X]\setminus\interior{A}{X}$ we have $x \in \closure{(\carrier[X]\setminus A)}{X}$. + \begin{subproof} + + Fix $x \in \carrier[X]\setminus\interior{A}{X}$. + Suppose not. + $x \notin \closure{(\carrier[X]\setminus A)}{X}$. + $x \in \carrier[X]$. + $(\carrier[X] \setminus \closure{(\carrier[X]\setminus A)}{X}) \inter \closure{(\carrier[X]\setminus A)}{X} = \emptyset$. + $x \in A$. + $x \in A \inter (\carrier[X] \setminus \closure{(\carrier[X]\setminus A)}{X})$. + $\carrier[X] \setminus \closure{(\carrier[X]\setminus A)}{X} \in \opens[X]$. + There exist $U \in \opens[X]$ such that $x \in U$ and $U\subseteq A$. + $U \subseteq \interior{A}{X}$. + Contradiction. + \end{subproof} + $\carrier[X]\setminus\interior{A}{X} \subseteq \closure{(\carrier[X]\setminus A)}{X}$. + $\closure{(\carrier[X]\setminus A)}{X} \subseteq \carrier[X]\setminus\interior{A}{X}$. \end{proof} + + \begin{definition}[Frontier]\label{frontier} $\frontier{A}{X} = \closure{A}{X}\setminus\interior{A}{X}$. \end{definition} +\begin{proposition}\label{closure_interior_frontier_is_in_carrier} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + Then $\closure{A}{X}, \interior{A}{X}, \frontier{A}{X} \subseteq \carrier[X]$. +\end{proposition} + +\begin{proposition}\label{frontier_is_closed} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. + Then $\frontier{A}{X}$ is closed in $X$. +\end{proposition} +\begin{proof} + $\closure{A}{X}\setminus\interior{A}{X}$ is closed in $X$ by \cref{closure_interior_frontier_is_in_carrier,closure_is_closed,interior_is_open,closed_minus_open_is_closed}. +\end{proof} + +\begin{proposition}\label{setdifference_eq_intersection_with_complement} + Suppose $A,B \subseteq C$. + Then $A \setminus B = A \inter (C \setminus B)$. +\end{proposition} -\begin{proposition}%[Frontier as intersection of closures] -\label{frontier_as_inter} + +\begin{proposition}\label{frontier_as_inter} + Let $X$ be a topological space. + Suppose $A \subseteq \carrier[X]$. $\frontier{A}{X} = \closure{A}{X} \inter \closure{(\carrier[X]\setminus A)}{X}$. \end{proposition} \begin{proof} - % TODO - Omitted. - %$\frontier{A}{X} = \closure{A}{X}\setminus\interior{A}{X}$. % by frontier (definition) - %$\closure{A}{X}\setminus\interior{A}{X} = \closure{A}{X}\inter(\carrier[X]\setminus\interior{A}{X})$. % by setminus_eq_inter_complement - %$\closure{A}{X}\inter(\carrier[X]\setminus\interior{A}{X}) = \closure{A}{X} \inter \closure{(\carrier[X]\setminus A)}{X}$. % by complement_interior_eq_closure_complement + \begin{align*} + \frontier{A}{X} \\ + &= \closure{A}{X}\setminus\interior{A}{X} \\ + &= \closure{A}{X} \inter (\carrier[X] \setminus \interior{A}{X}) \explanation{by \cref{setdifference_eq_intersection_with_complement,closure_interior_frontier_is_in_carrier}}\\ + &= \closure{A}{X} \inter \closure{(\carrier[X]\setminus A)}{X} \explanation{by \cref{complement_interior_eq_closure_complement}} + \end{align*} \end{proof} \begin{proposition}\label{frontier_of_emptyset} @@ -264,3 +529,7 @@ \begin{definition}\label{neighbourhoods} $\neighbourhoods{x}{X} = \{N\in\pow{\carrier[X]} \mid \exists U\in\opens[X]. x\in U\subseteq N\}$. \end{definition} + +\begin{definition}\label{neighbourhoods_set} + $\neighbourhoodsSet{x}{X} = \{N\in\pow{\carrier[X]} \mid \exists U\in\opens[X]. x\subseteq U\subseteq N\}$. +\end{definition} |