779 \end{figure}

   779 \end{figure}

   780 

   780 

   781 

   781 

   782 \begin{axiom}[Splittings]

   782 \begin{axiom}[Splittings]

   783 \label{axiom:vcones}

   783 \label{axiom:vcones}

   785 let $P$ be a finite poset of splittings of $c$.

   785 let $P$ be a finite poset of splittings of $c$.

   786 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.

   786 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.

   787 Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation.

   787 Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation.

   788 Also, any splitting of $\bd c$ can be extended to a splitting of $c$.

   788 Also, any splitting of $\bd c$ can be extended to a splitting of $c$.

   789 \end{axiom}

   789 \end{axiom}