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\import{topology/topological-space.tex}
% T0 separation
\begin{definition}\label{is_kolmogorov}
$X$ is Kolmogorov iff
for all $x,y\in\carrier[X]$ such that $x\neq y$
there exist $U\in\opens[X]$ such that
$x\in U\not\ni y$ or $x\notin U\ni y$.
\end{definition}
\begin{abbreviation}\label{kolmogorov_space}
$X$ is a Kolmogorov space iff $X$ is a topological space and
$X$ is Kolmogorov.
\end{abbreviation}
\begin{abbreviation}\label{teezero}
$X$ is \teezero\ iff $X$ is Kolmogorov.
\end{abbreviation}
\begin{abbreviation}\label{teezero_space}
$X$ is a \teezero-space iff $X$ is a Kolmogorov space.
\end{abbreviation}
\begin{proposition}\label{kolmogorov_implies_kolmogorov_for_closeds}
Suppose $X$ is a Kolmogorov space.
Let $x,y\in\carrier[X]$.
Suppose $x\neq y$.
Then there exist $A\in\closeds{X}$ such that
$x\in A\not\ni y$ or $x\notin A\ni y$.
\end{proposition}
\begin{proof}
Take $U\in\opens[X]$ such that $x\in U\not\ni y$ or $x\notin U\ni y$
by \cref{is_kolmogorov}.
Then $\carrier[X]\setminus U\in\closeds{X}$ by \cref{complement_of_open_elem_closeds}.
Now $x\in (\carrier[X]\setminus U)\not\ni y$ or $x\notin (\carrier[X]\setminus U)\ni y$
by \cref{setminus}.
\end{proof}
\begin{proposition}\label{kolmogorov_for_closeds_implies_kolmogorov}
Suppose for all $x,y\in\carrier[X]$ such that $x\neq y$
there exist $U\in\closeds{X}$ such that
$x\in U\not\ni y$ or $x\notin U\ni y$.
Then $X$ is Kolmogorov.
\end{proposition}
\begin{proof}
Follows by \cref{closeds,is_closed_in,is_kolmogorov,setminus}.
\end{proof}
\begin{proposition}\label{kolmogorov_iff_kolmogorov_for_closeds}
Let $X$ be a topological space.
$X$ is Kolmogorov iff
for all $x,y\in\carrier[X]$ such that $x\neq y$
there exist $U\in\closeds{X}$ such that
$x\in U\not\ni y$ or $x\notin U\ni y$.
\end{proposition}
\begin{proof}
Follows by \cref{kolmogorov_implies_kolmogorov_for_closeds,kolmogorov_for_closeds_implies_kolmogorov}.
\end{proof}
% T1 separation (Fréchet topology)
\begin{definition}\label{teeone}
$X$ is \teeone\ iff
for all $x,y\in\carrier[X]$ such that $x\neq y$
there exist $U, V\in\opens[X]$ such that
$U\ni x\notin V$ and $V\ni y\notin U$.
\end{definition}
\begin{abbreviation}\label{teeone_space}
$X$ is a \teeone-space iff $X$ is a topological space and
$X$ is \teeone.
\end{abbreviation}
\begin{proposition}\label{teeone_implies_singletons_closed}
Let $X$ be a \teeone-space.
Assume $x \in \carrier[X]$.
Then $\{x\}$ is closed in $X$.
%Then for all $x\in\carrier[X]$ we have $\{x\}$ is closed in $X$. Ambigus Phrase. if Omitted is ereased.
\end{proposition}
\begin{proof}
%Omitted.
%Fix $x \in X$.
Let $V = \{ U \in \opens[X] \mid x \notin U\}$.
For all $y \in \carrier[X]$ such that $x \neq y$ there exist $U \in \opens[X]$ such that $x \notin U \ni y$.
For all $y \in \carrier[X]$ such that $y \neq x$ there exists $U \in V$ such that $y \in U$.
$\unions{V} \in \opens[X]$.
For all $y \in \carrier[X]$ such that $x \neq y$ we have $y \in \unions{V}$.
We show that $\carrier[X] \setminus \{x\} = \unions{V}$.
\begin{subproof}
We show that for all $y \in \carrier[X] \setminus \{x\}$ we have $y \in \unions{V}$.
\begin{subproof}
Fix $y \in \carrier[X] \setminus \{x\}$.
$y \neq x$.
$y \in \carrier[X]$.
$y \in \unions{V}$.
\end{subproof}
For all $y \in \unions{V}$ we have $y \notin \{x\}$.
For all $y \in \unions{V}$ we have $y \in \carrier[X] \setminus \{x\}$.
Follows by set extensionality.
\end{subproof}
% Let $U_{y} \in \opens[X]$ such that $x \neq y \in X$. Error: unexpected '{' expecting digit
%For all $y$ there exists $U$ such that $x \neq y$ and $y \in U$ and $U \in \opens[X]$.
% TODO
% Choose for every y distinct from x and open subset U_y containing y but not x.
% The union U of all the U_y is open.
% {x} is the complement of U in \carrier[X].
\end{proof}
%
% Conversely, if \{x\} is open, then for any y distinct from x we can use
% X\setminus\{x\} as the open neighbourhood of y.
% T2 separation
\begin{definition}\label{is_hausdorff}
$X$ is Hausdorff iff
for all $x,y\in\carrier[X]$ such that $x\neq y$
there exist $U, V\in\opens[X]$ such that
$x\in U$ and $y\in V$ and $U$ is disjoint from $V$.
\end{definition}
\begin{abbreviation}\label{hausdorff_space}
$X$ is a Hausdorff space iff $X$ is a topological space and
$X$ is Hausdorff.
\end{abbreviation}
\begin{abbreviation}\label{teetwo}
$X$ is \teetwo\ iff $X$ is Hausdorff.
\end{abbreviation}
\begin{abbreviation}\label{teetwo_space}
$X$ is a \teetwo-space iff $X$ is a Hausdorff space.
\end{abbreviation}
\begin{proposition}\label{teeone_space_is_teezero_space}
Let $X$ be a \teeone-space.
Then $X$ is a \teezero-space.
\end{proposition}
\begin{proof}
Follows by \cref{is_kolmogorov,teeone}.
\end{proof}
\begin{proposition}\label{teetwo_space_is_teeone_space}
Let $X$ be a \teetwo-space.
Then $X$ is a \teeone-space.
\end{proposition}
\begin{proof}
We show that for all $x,y\in\carrier[X]$ such that $x\neq y$
there exist $U, V\in\opens[X]$ such that
$U\ni x\notin V$ and $V\ni y\notin U$.
\begin{subproof}
$X$ is hausdorff.
For all $x,y\in\carrier[X]$ such that $x\neq y$
there exist $U, V\in\opens[X]$ such that
$x\in U$ and $y\in V$ and $U$ is disjoint from $V$.
\end{subproof}
\end{proof}
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