\chapter{Countable Features of Spaces: Size Restrictions}

\begin{exercise}
Show that $A$ is dense in $X$ if and only if every non-empty open set of $X$ contains a point of $A$.
\end{exercise}

\oneperpage

\begin{exercise}
Show that $\R_{\textrm{std}}$ is separable. 
With which of the topologies on $\R$ that you have studied is $\R$ not separable?
\end{exercise}

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\begin{exercise}
Add 'separable' as a new property in your chart, and complete your chart by deciding which of the spaces we've studied are separable.
\end{exercise}

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\begin{exercise}
Find a separable space that contains a subspace that is not separable in the subspace topology.
\end{exercise}

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\begin{thm}    
If $X$ and $Y$ are separable spaces, then $X\times Y$ is separable. 
\end{thm}

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\begin{thm}
The space $2^\R$ is separable.
\end{thm}

\oneperpage

\begin{exercise}     Let $\{ X_\beta\}_{\beta \in\mu}$ be a collection
of separable spaces where $\vert\mu\vert \le 2^{\omega_0}$, then 
$\prod_{\beta \in\mu} X_\beta$ is separable.
   \begin{inst}
      A special case of the HewittÐMarczewskiÐPondiczery theorem.
   \end{inst}
\end{exercise}

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\begin{exercise}     
If $X$ is a separable, Hausdorff space, then
$| X | \le \left| 2^{2^\N}\right|$.
  
\end{exercise}

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\begin{thm}    
Let $X$ be a $2^{\textrm{nd}}$ countable space, then $X$ is separable.
\end{thm}

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\begin{exercise}
\label{examples-2nd-countable-separable}
\begin{enumerate}
\item The space $\R_{\textrm{std}}$ is $2^{\text{nd}}$ countable (and hence separable).
\item The space $\R_{\textrm{LL}}$ is separable but not $2^{\text{nd}}$ countable.
\item The space $\Hbub$ is separable but not $2^{\text{nd}}$ countable.
\end{enumerate}
\end{exercise}

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\begin{thm}     
Every uncountable set in a $2^{\textrm{nd}}$ countable space
has a limit point.
\end{thm}

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\begin{exercise}   
A $2^{\textrm{nd}}$ countable space is hereditarily $2^{\textrm{nd}}$ countable.
\end{exercise}

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\begin{exercise}
If $X$ and $Y$ are 2nd countable spaces, then $X\times Y$ is 2nd countable.
\end{exercise}

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\begin{thm}    
Let $X$ be a $2^{\textrm{nd}}$ countable space. Then $X$ is $1^{\textrm{st}}$ countable.
\end{thm}

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\begin{thm}    
If $X$ is a topological space, $p\in X$, and $p$ has a countable neighborhood basis, then $p$ has a nested countable neighborhood basis.
\end{thm}

\oneperpage

\begin{exercise}
\begin{enumerate}
\item The space $\R_\textrm{LL}$ is $1^{\textrm{st}}$ countable.
\item The space $\Hbub$ is $1^{\textrm{st}}$ countable.
\item The space $2^\R$ is not $1^{\textrm{st}}$ countable.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
You may as well extend your table of spaces and properties by adding new rows for the properties $1^{\textrm{st}}$ countable and $2^{\textrm{nd}}$ countable and determining those properties for each of your spaces.
\end{exercise}

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\begin{thm}     
Suppose $x$ is a limit point of the set $A$ in a $1^{\textrm{st}}$ countable space $X$.  Then there is a sequence of points $\{ a_i\}_{i \in \N}$  in $A$ that converges to $x$.
\end{thm}

\oneperpage

\begin{exercise}   
A $1^{\textrm{st}}$ countable space is hereditarily $1^{\textrm{st}}$ countable.
\end{exercise}

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\begin{exercise}
If $X$ and $Y$ are 1st countable spaces, then $X\times Y$ is 1st countable.
\end{exercise}

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\begin{exercise}
Show that the real line with the standard topology is Souslin.
\end{exercise}

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\begin{thm}     
A separable space has the Souslin property.
\end{thm}

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\begin{thm}
\label{thm:2^X is souslin}
For any set $X$, the topological space $2^X$ has the Souslin property.
 \end{thm}

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\begin{exercise}
Find a Souslin space that is not separable.
\end{exercise}

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\begin{thm*}    
Let $\{ X_\beta\}_{\beta \in\mu}$ be a collection of separable spaces, then $\prod_{\beta \in\mu} X_\beta$ is Souslin.
\end{thm*}

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