\chapter{Separation Properties: Separating This from That}

\begin{thm}
A space $(X,\Tau)$ is $T_1$ if and only if every point in $X$ is a closed set.
\end{thm}

\oneperpage

\begin{exercise}
Let $X$ be a space with the finite complement topology.  Show that $X$ is $T_1$.
\end{exercise}

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\begin{exercise}
Show that $\R_\mathrm{std}$ is Hausdorff.
\end{exercise}

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\begin{exercise}
Show that $\Hbub$ is regular.
\end{exercise}

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\begin{exercise}
Show that $\RLL$ is normal.
\end{exercise}

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\begin{exercise} 
  \begin{inst}
    You should definitely do this problem if you aren't doing the chapter on metric spaces.  The ideas here are helpful in proving that metric spaces are normal.
   \end{inst}
\begin{enumerate} 
\item Consider $\R^2$ with the standard topology. Let $p \in \R^2$ be a point not in a closed set $A$. Show that $\inf \{ d(a,p)\mid a\in A\} >0$.  (Recall that $\inf E$ is the greatest lower bound of a set of real numbers $E$.)
\item Show that $\R^2$ with the standard topology is regular.
\item Find two disjoint closed subsets $A$ and $B$ of $\R^2$ with the standard topology such that  
$$\inf \{ d(a,b)\mid a\in A \text{ and }b\in B\} =0.$$
\item Show that $\R^2$ with the standard topology is normal.
\end{enumerate}
\end{exercise}

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\begin{thm}
\begin{enumerate}
%\item A $T_1$-space is a $T_0$-space.
\item A $T_2$-space (Hausdorff) is a $T_1$-space.
\item A $T_3$-space (regular and $T_1$) is a Hausdorff space, that is, a $T_2$-space.
\item A $T_4$-space (normal and $T_1$) is regular and $T_1$, that is, a $T_3$-space.
\end{enumerate}
\end{thm}

\oneperpage

\begin{thm}
\label{thm:regular-alt}
A topological space $X$ is regular if and only
if for each point $p$ in $X$ and open set $U$ containing $p$ there exists an
open set $V$ such that $p\in V$ and $\overline{V} \subset U$.
\end{thm}

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\begin{thm}
\label{thm:normal-alt}    
A topological space $X$ is normal if and only
if for each closed set $A$ in $X$ and open set $U$ containing $A$ there
exists an open set $V$ such that $A\subset V$, and $\overline{V} \subset U$.
\end{thm}

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\begin{thm}    
A topological space $X$ is normal if and only if
for each pair of disjoint closed sets $A$ and $B$, there are disjoint
open sets $U$ and $V$ such that $A\subset U$, $B\subset V$, and
$\overline{U} \cap \overline{V} = \emptyset$.
\end{thm}

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\begin{thm}[The Incredible Shrinking Theorem]
\label{incredible-shrinking-theorem}
A topological space $X$ is normal if and only if
for each pair of open sets $U$, $V$ such that $U \cup V = X$, there exist open sets $U'$, $V'$ such that 
$\ol{U'} \subset U$ and $\ol{V'} \subset V$, and $U' \cup V' = X$.
\end{thm}

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\begin{exercise}  
\begin{enumerate}
%\item Describe an example of a topological space that is $T_0$ but not $T_1$. 
\item Describe an example of a topological space that is $T_1$ but not $T_2$. 
\item Describe an example of a topological space that is $T_2$ but not $T_3$. 
\item Describe an example of a topological space that is $T_3$ but not $T_4$. 
\end{enumerate}
\end{exercise}

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\begin{exercise}
Construct a table, listing our previous examples of topological spaces as column titles, and listing the separation properties as row titles. In each box, answer the question of whether the example of the column has the property of the row. Here are the spaces to use as column titles:
\begin{enumerate}
\item $\R_{std}$
\item $\R^n_{std}$
\item indiscrete topology
\item discrete topology
\item finite complement topology
\item countable complement topology
\item lower limit topology, $\RLL$
\item double headed snake, $\R_{+00}$
\item $\R_{har}$
\item Sticky Bubble Topology, $\Hbub$
\item arithmetic progression topology, $\Z_{arith}$
\item lexicographically ordered square
\item $2^X$
\end{enumerate}
Here are the properties to use as row titles:
\begin{enumerate}
\item $T_1$
\item Hausdorff
\item regular
\item normal
\end{enumerate}
\end{exercise}

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\begin{exercise}
Show that $\Hbub$ is not normal.
\end{exercise}

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\begin{thm}
Order topologies are $T_1$, Hausdorff, regular, and normal.
\end{thm}

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\begin{thm}    
Let $X$ and $Y$ be Hausdorff.  Then $X \times Y$ is Hausdorff.
\end{thm}

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\begin{thm}
Let $X$ and $Y$ be regular.  Then $X \times Y$ is regular.
   
\end{thm}

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\begin{exercise}
Show that $\RLL \times \RLL$ is not normal.  It may help to consider the ``negative diagonal'' line $L$.
  
\end{exercise}

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\begin{thm} 
\label{hereditary-Hausdorff}   
Every Hausdorff space is hereditarily Hausdorff.
\end{thm}

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\begin{thm}    
Every regular space is hereditarily regular.
\end{thm}

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\begin{exercise}
\begin{enumerate} 
\item Prove that the space $2^\R$ is normal.
\item Prove that if you remove a single point from $2^\R$, the resulting subspace is not normal. 
\end{enumerate}
\end{exercise}

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\begin{exercise}(Walking the Tychonoff Plank, or Mutiny on the Boundary)
\begin{enumerate}
\item Show that the Tychonoff Plank is normal.
\item Show that the Tychonoff Plank minus the single point $(\omega_0, \omega_1)$ is not normal.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}   
\label{thm:normal-closed}
Let $A$ be a closed subset of a normal space
$X$.  Then $A$ is normal when given the relative topology.
  \begin{inst}
    If you skip this section, you might assign this problem as an exercise.  
    It is the most important theorem from this section.
  \end{inst}
\end{thm}

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\begin{exercise}
\begin{enumerate}
\item Prove that for any set $X$, $2^X$ is normal. (This part is not really different from showing that $2^{\R}$ is normal, which you did in a previous exercise.)
\item Recall that there is a one-to-one correspondence between the points of $2^X$ and subsets of $X$, as follows: recall that each point of $2^X$ is a function $f:X \to \{0,1\}$, so $f^{-1}(1)$ is a subset of $X$. Let $C \subset 2^X$ consist of those points that take on the value $1$ on only a countable set of coordinates, that is, $C$ is the set of functions $f:X \to \{0,1\}$, for which $f^{-1}(1)$ is countable. Prove that $C$ with the subspace topology is normal. 
\end{enumerate}
\end{exercise}

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\begin{exercise}
Let $Y$ be a subspace of a topological space $X$, and let $A$ and $B$ be two disjoint closed subsets of $Y$ in the subspace topology.  Show that both $\ol{A} \cap B = \emptyset$ and $A \cap \ol{B} = \emptyset$, where the closures are taken in $X$.
\end{exercise}

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\begin{thm}
The space $X$ is a completely normal space if and only if $X$ is hereditarily normal.
\end{thm}

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\begin{exercise}
\begin{enumerate}
\item
Recall that $\R$ is an order topology.  Find a subset of $\R$ where the subspace topology is not the order topology on the subset.
\item
Find a line in the lexicographically ordered square whose relative topology is the discrete topology on this line, but this is not the order topology on the subset.
\item 
Notice that $\R_{LL}$ is not an order topology.  Find a line in the lexicographically ordered square whose relative topology is the lower limit topology.
\end{enumerate}
\end{exercise}

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\begin{thm}
Order topologies are hereditarily normal.
\end{thm}

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\begin{thm}[The Normality Lemma]
\index{Normality Lemma}
%\begin{normlemma}
\label{thm:normality-lemma}
Let $A$ and $B$ be subsets of a 
topological space $X$ and let $\{ U_i\}_{i\in \N}$ and
$\{ V_i\}_{i\in\N}$ be two collections of open sets such that
\begin{enumerate}
\item $A\subset \bigcup_{i\in\N} U_i$,
\item $B\subset \bigcup_{i\in\N} V_i$,
\item for each $i$ in $\N$, $\overline{U}_i \cap B=\emptyset$
and $\overline{V}_i \cap A = \emptyset$.
\end{enumerate}
Then there exist open sets $U$ and $V$ such that $A\subset U$,
$B\subset V$, and $U\cap V=\emptyset$.
%\end{normlemma}
\end{thm}

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\begin{thm}
If $X$ is normal and $C = \cup_{i\in\N}  K_i$ is the union of closed sets $K_i$ in $X$, then the subspace $C$ is normal.
\end{thm}

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\begin{thm}
Suppose a space $X$ is regular and countable. Then $X$ is normal.
\end{thm}

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\begin{thm}
Suppose a space $X$ is regular and has a countable basis. Then $X$ is normal.
\end{thm}

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\begin{thm}
Let $X$ be a perfectly normal space. Then $X$ is completely normal.
\end{thm}

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