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\import{set.tex}
\import{set/cons.tex}
\import{set/powerset.tex}
\import{set/fixpoint.tex}
\import{set/product.tex}
\import{topology/topological-space.tex}
\import{topology/separation.tex}
\import{topology/continuous.tex}
\import{topology/basis.tex}
\import{numbers.tex}
\import{function.tex}
\section{The canonical topology on $\mathbbR$}
\begin{definition}\label{topological_basis_reals_eps_ball}
$\topoBasisReals = \{ \epsBall{x}{\epsilon} \mid x \in \reals, \epsilon \in \realsplus\}$.
\end{definition}
\begin{axiom}\label{reals_carrier_reals}
$\carrier[\reals] = \reals$.
\end{axiom}
\begin{lemma}\label{intervals_are_connected_in_reals}
Suppose $a,b \in \reals$.
Then for all $c \in \reals$ such that $a < c < b$ we have $c \in \intervalopen{a}{b}$.
\end{lemma}
\begin{lemma}\label{epsball_are_subset_reals_elem}
Suppose $x \in \reals$.
Suppose $\epsilon \in \realsplus$.
Then for all $y \in \epsBall{x}{\epsilon}$ we have $y \in \reals$.
\end{lemma}
\begin{lemma}\label{intervalopen_iff}
Suppose $a,b,c \in \reals$.
Suppose $a < b$.
$c \in \intervalopen{a}{b}$ iff $a < c < b$.
\end{lemma}
\begin{lemma}\label{epsball_are_subseteq_reals_set}
Suppose $x \in \reals$.
Suppose $\epsilon \in \realsplus$.
Then $\epsBall{x}{\epsilon} \subseteq \reals$.
\end{lemma}
\begin{lemma}\label{epsball_are_subset_reals_set}
Suppose $x \in \reals$.
Suppose $\epsilon \in \realsplus$.
Then $\epsBall{x}{\epsilon} \subset \reals$.
\end{lemma}
\begin{lemma}\label{reals_order_minus_positiv}
Suppose $x,y \in \reals$.
Suppose $\zero < y$.
$x - y < x$.
\end{lemma}
\begin{lemma}\label{realsplus_bigger_zero}
For all $x \in \realsplus$ we have $\zero < x$.
\end{lemma}
\begin{lemma}\label{realsplus_in_reals}
For all $x \in \realsplus$ we have $x \in \reals$.
\end{lemma}
\begin{lemma}\label{epsball_are_inhabited}
Suppose $x \in \reals$.
Suppose $\epsilon \in \realsplus$.
Then $\epsBall{x}{\epsilon}$ is inhabited.
\end{lemma}
\begin{proof}
$x < x + \epsilon$.
$x - \epsilon < x$.
$x \in \epsBall{x}{\epsilon}$.
\end{proof}
\begin{lemma}\label{reals_elem_inbetween}
For all $a,b \in \reals$ such that $a < b$ we have there exists $c \in \reals$ such that $a < c < b$.
\end{lemma}
\begin{lemma}\label{epsball_equal_openinterval}
Suppose $x \in \reals$.
Suppose $\epsilon \in \realsplus$.
Then $\epsBall{x}{\epsilon} = \intervalopen{x - \epsilon}{x + \epsilon}$.
\end{lemma}
\begin{lemma}\label{minus_behavior1}
For all $x \in \reals$ we have $x - x = \zero$.
\end{lemma}
\begin{lemma}\label{minus_behavior2}
For all $x \in \reals$ we have $x + \neg{x} = \zero$.
\end{lemma}
\begin{lemma}\label{minus_behavior3}
For all $x \in \reals$ we have $\neg{x} = \zero - x$.
\end{lemma}
\begin{lemma}\label{reals_order_is_addition_with_positiv_number}
For all $x,y \in \reals$ such that $x < y$ we have there exists $z \in \realsplus$ such that $x + z = y$.
\end{lemma}
\begin{proof}
%Fix $x,y \in \reals$.
\end{proof}
\begin{lemma}\label{reals_order_is_transitive}
For all $x,y,z \in \reals$ such that $x < y$ and $y < z$ we have $x < z$.
\end{lemma}
\begin{lemma}\label{reals_order_plus_minus}
Suppose $a,b \in \reals$.
Suppose $\zero < b$.
Then $(a-b) < (a+b)$.
\end{lemma}
\begin{proof}
We show that $a < (a+b)$.
\begin{subproof}
Trivial.
\end{subproof}
We show that $(a-b) < a$.
\begin{subproof}
Trivial.
\end{subproof}
\end{proof}
\begin{lemma}\label{epsball_are_connected_in_reals}
Suppose $x \in \reals$.
Suppose $\epsilon \in \realsplus$.
Then for all $c \in \reals$ such that $(x - \epsilon) < c < (x + \epsilon)$ we have $c \in \epsBall{x}{\epsilon}$.
\end{lemma}
\begin{proof}
$x - \epsilon \in \reals$.
$x + \epsilon \in \reals$.
It suffices to show that for all $c$ such that $c \in \reals \land (x - \epsilon) < c < (x + \epsilon)$ we have $c \in \epsBall{x}{\epsilon}$.
%Fix $c$ such that $c \in \reals \land (x - \epsilon) < c < (x + \epsilon)$.
%Suppose $(x - \epsilon) < c < (x + \epsilon)$.
\end{proof}
\begin{theorem}\label{topological_basis_reals_is_prebasis}
$\topoBasisReals$ is a topological prebasis for $\reals$.
\end{theorem}
\begin{proof}
We show that $\unions{\topoBasisReals} \subseteq \reals$.
\begin{subproof}
It suffices to show that for all $x \in \unions{\topoBasisReals}$ we have $x \in \reals$.
Fix $x \in \unions{\topoBasisReals}$.
\begin{byCase}
\caseOf{$x = \emptyset$.}
Trivial.
\caseOf{$x \neq \emptyset$.}
There exists $U \in \topoBasisReals$ such that $x \in \topoBasisReals$.
Take $U \in \topoBasisReals$ such that $x \in \topoBasisReals$.
\end{byCase}
\end{subproof}
We show that $\reals \subseteq \unions{\topoBasisReals}$.
\begin{subproof}
It suffices to show that for all $x \in \reals$ we have $x \in \unions{\topoBasisReals}$.
Fix $x \in \reals$.
$\epsBall{x}{1} \in \topoBasisReals$.
Therefore $x \in \unions{\topoBasisReals}$.
\end{subproof}
\end{proof}
\begin{theorem}\label{topological_basis_reals_is_basis}
$\topoBasisReals$ is a topological basis for $\reals$.
\end{theorem}
\begin{proof}
$\topoBasisReals$ is a topological prebasis for $\reals$ by \cref{topological_basis_reals_is_prebasis}.
Let $B = \topoBasisReals$.
It suffices to show that for all $U \in B$ we have for all $V \in B$ we have for all $x$ such that $x \in U, V$ there exists $W\in B$ such that $x\in W\subseteq U, V$.
Fix $U \in B$.
Fix $V \in B$.
It suffices to show that for all $x \in U \inter V$ there exists $W\in B$ such that $x\in W\subseteq U, V$.
Fix $x \in U \inter V$.
\begin{byCase}
\caseOf{$U \inter V = \emptyset$.}
Trivial.
\caseOf{$U \inter V \neq \emptyset$.}
Then $U \inter V$ is inhabited.
%It suffices to show that
\end{byCase}
\end{proof}
\begin{axiom}\label{topological_space_reals}
$\opens[\reals] = \genOpens{\topoBasisReals}{\reals}$.
\end{axiom}
\begin{theorem}\label{reals_is_topological_space}
$\reals$ is a topological space.
\end{theorem}
\begin{proof}
$\topoBasisReals$ is a topological basis for $\reals$.
Let $B = \topoBasisReals$.
We show that $\opens[\reals]$ is a family of subsets of $\carrier[\reals]$.
\begin{subproof}
It suffices to show that for all $A \in \opens[\reals]$ we have $A \subseteq \reals$.
Fix $A \in \opens[\reals]$.
Follows by \cref{powerset_elim,topological_space_reals,genopens}.
\end{subproof}
We show that $\reals \in\opens[\reals]$.
\begin{subproof}
$B$ covers $\reals$ by \cref{topological_prebasis_iff_covering_family,topological_basis}.
$\unions{B} \in \genOpens{B}{\reals}$.
$\reals \subseteq \unions{B}$.
\end{subproof}
We show that for all $A, B\in \opens[\reals]$ we have $A\inter B\in\opens[\reals]$.
\begin{subproof}
Follows by \cref{topological_space_reals,inters_in_genopens}.
\end{subproof}
We show that for all $F\subseteq \opens[\reals]$ we have $\unions{F}\in\opens[\reals]$.
\begin{subproof}
Follows by \cref{topological_space_reals,union_in_genopens}.
\end{subproof}
$\carrier[\reals] = \reals$.
Follows by \cref{topological_space}.
\end{proof}
\begin{proposition}\label{open_interval_is_open}
Suppose $a,b \in \reals$.
Then $\intervalopen{a}{b} \in \opens[\reals]$.
\end{proposition}
\begin{lemma}\label{safetwo}
Contradiction.
\end{lemma}
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