:
\begin{enumerate}
\item $\Aaa$ is disjointifiable,
\item  $K_\Aaa$ is an  Eberlein compact,
\item $C(K_\Aaa)$ is isomorphic to $c_0(\kappa)$ for $\kappa=|\Aaa|+\omega$,
\item $C(K_\Aaa)$ has an unconditional basis,
\item $C(K_\Aaa)$ has a projectional skeleton.
\end{enumerate}
\end{tw}
\begin{proof}Apply \ref{eberleindisjoint}, \ref{eberleinc0}, \ref{eberleinandskeletons}.
\end{proof}

In Section 3 we consider the case corresponding to Banach spaces which are
Lindel\"of in the weak topology. We obtain the following:


\begin{tw}\label{monolithiclist} Suppose $\Aaa$ is an almost disjoint family of countable sets, then the following are equivalent
\begin{enumerate}
\item $\Aaa$ is locally small,
\item  $K_\Aaa$ is monolithic,
\item $C(K_\Aaa)$ has the controlled SCP,
\item $C(K_\Aaa)$ is Lindel\"of in the weak topology,
\item $B_{C(K_\Aaa)^*}$ is monolithic  in the weak$^*$ topology.
\end{enumerate}
\end{tw}
\begin{proof}Apply \ref{equivwlindelof} and \ref{equivmonolithic}.
\end{proof}

A locally small almost disjoint family is defined in \ref{locsmalldef},
a compact space $K$ is \emph{monolithic} if all of its separable subsets are second countable.
In case $K$ is scattered, this just means that separable subsets are countable.
The notion of a monolithic space is due to Arkhangel'skii (see~\cite{Ark_book}).
A monolithic space is often called \emph{$\omega$-monolithic}, since this notion clearly generalizes to higher cardinals.
Typical examples of monolithic spaces are \emph{Corson compacta}, namely, compact subsets of $\Sigma$-products $\Sigma(\Gamma)$, where $\Sigma(\Gamma)$ is the subspace of $\Err^\Gamma$ consisting of all functions $\map x \Gamma \Err$ whose support $\supp(x) = \setof{\al}{x(\al)\ne0}$ is countable.
It is obvious that the closure of every countable subset of $\Sigma(\Gamma)$ is compact and second countable (contained in $\Err^S$ for some countable $S \subs \Gamma$).

The above results and  our examples \ref{nonmono} and \ref{nonprojskeleton} 
decide completely which implications hold among the properties of Definition \ref{maindefinition}
namely (1)$\Leftrightarrow$(3), (4)$\Rightarrow $(3), (2)$\Rightarrow$(1) and (4)$\Rightarrow$(2).
The last implication follows from a well known fact that Eberlein compacta are monolithic, being Corson compact.

In general the controlled SCP does not imply the Lindel\"of property
in the weak topology. To see this, take any non-metrizable monolithic compact linearly ordered space $K$ (e.g., $K$ could be a compact Aronszajn line) and
apply Nakhmanson's
theorem \cite{nakhmanson}, which says that for a linearly ordered compact $K$ the Lindel\"of
property of $C(K)$ implies that $K$ is second countable; by~\cite[Thm. 1.6]{KalKub2}, $C(K)$ has the controlled SCP.
It is interesting that a sufficient condition given in~\cite[Thm. 12.33]{fabianetal} for the Lindel\"of property due to Orihuela has a flavor of the controlled SCP.



In Section 4 we obtain some structural results concerning locally small almost disjoint families of countable sets,
of cardinality $\omega_1$, to express them  we need the notion of a ladder system (see the beginning of Section 4)
and we need to introduce some simple definition: Suppose $\Aaa\cup\Bee$ is an almost disjoint family 
$\Aaa'\subseteq \Aaa$ and  $\phi: \Aaa'\rightarrow
\Bee$ is an injection. Then 
$$(\Aaa\setminus \Aaa')\cup\{A'\cup \phi(A'): A'\in \Aaa'\}\cup(\Bee\setminus \phi[\Aaa'])$$
 is denoted $\Aaa+_\phi\Bee$.
The main result of Section 4  is the following structural theorem:

\begin{tw}\label{structural} Suppose that $\Aaa$ is a locally small  almost disjoint family of countable
sets of cardinality $\omega_1$. Then $\Aaa$ is equivalent to the family of the form
$$\bigcup_{n\in \Nat}(\Aaa_n+_{\phi_n}\Bee_n)$$
where
all $\Aaa_n$s are ladder systems, all $\Bee_n$s are disjointifiable
and $\phi_n$s are injections defined  on subfamilies of $\Aaa_n$s.
\end{tw}

In other words, locally small almost disjoint families are obtained in a canonical way from countably many ladder systems and pairwise disjoint families. As noted at the end of Section 4, if all the families
$\Aaa_n$ are disjointifiable the $\Aaa$ is disjointifiable as well.
One should note that the structure of non locally small almost disjoint families
of sets, even in the case of  subsets of $\Nat$ is a much harder task with extensive literature devoted to it.


Analyzing Banach spaces induced by the ladder system spaces in Section 4, we obtain the following

\begin{tw}\label{nonprojskeleton} There is an almost disjoint family $\Aaa$ such that 
$C(K_\Aaa)$  has controlled SCP and continuous SCP but it has neither a PRI nor a projectional skeleton.
\end{tw}


Actually the above result provides the first example of  a Banach space with the continuous SCP, not isomorphic to a space with a PRI. This question was raised in \cite{Kub_lines} and was motivated by
an observation that in a Banach space of density $\aleph_1$, a skeleton of $1$-complemented separable subspaces naturally leads to a PRI. 
We should  note here that in general the existence of a PRI in the spaces $C(K_\Aaa)$ does 
not imply even the SCP:

\begin{ex}
Let $\Bee \subs [\omega]^\omega$ be an almost disjoint family of size $\omega_1$ and let $\Aaa \subs [\omega_2 \setminus \omega]^\omega$ be a disjoint family of size $\omega_2$.
Let $\Cee = \Aaa \cup \Bee$.
The space $C(K_\Bee)$ is a complemented subspace of $C(K_\Cee)$ and the canonical copy of $c_0$ is not complemented in $C(K_\Bee)$ (see the proof of Proposition~\ref{uncomplemented} for details).
Thus, $C(K_\Cee)$ fails the SCP.
On the other hand, it is easy to construct a PRI on $C(K_\Cee)$.
In fact, let $\sett{P_\al}{\al < \omega_2}$ be a PRI on $C(K_\Aaa)$ and, knowing that $C(K_\Cee) = C(K_\Aaa) \oplus C(K_\Bee)$, extend
$P_{\omega_1}$ by enlarging its kernel to $\ker{P_{\omega_1}} \oplus C(K_\Bee)$. Finally, replace each $P_\al$ for $\al > \omega_1$ by the appropriate projection onto $\im P_\al \oplus C(K_\Bee)$.
\end{ex}

Having in mind the results of Section 3, the only property from \ref{maindefinition} which can hold for
non-monolithic $K_\Aaa$s is the SCP (and equivalent continuous SCP). It turns out that 
such an example exists, its construction is the subject matter of Section 6: 

\begin{tw}\label{nonmono} There is an almost disjoint family $\Aaa$ which is not locally small, i.e.,
none of the conditions of \ref{monolithiclist} like the controlled SCP are satisfied by $C(K_\Aaa)$, yet
the space $C(K_\Aaa)$  has the SCP.
\end{tw}


This is a different behavior than for linearly ordered compact spaces $K$, as it was proved