Orderability of topological spaces with a connected linear base

Let $\gamma$ be a system of subsets in a set $X$. A string of elements of $\gamma$ is denned to be any finite subsystem $\delta=\{U_1,U_2,\dots,U_n\}$ such that $U_i\cap U_{i+1}\ne\emptyset$ for all $i=1,2,\dots,n-1$. If $a$ and $b$, $a\ne b$, are points in $X$, then the $\gamma$-ray from $a$ to $b$ is defined as the set $\gamma L(a,b)=\{x\in X:{}$there is a string $\delta\subset\gamma$ and $\{a,x\}\subset\cup\delta\not\ni b\}$. A base $\gamma$ of a topological space $X$ is called connected if $b\in[\gamma L(a,b)]$ for all $a\ne b$ из $X$ in $X$.
In the article we prove that every base of a connected $T_1$-space is connected.
We call a system $\mathcal{B}$ linear if it satisfies the following conditions:
$\mathcal{PB}$. If three sets in the system $\mathcal{B}$ are such that every two of them meet one another then there is a pair of them with the intersection lying in the third set.
$\mathcal{UB}$. If two sets in the system $\mathcal{B}$ meet one another then their union belongs to $\mathcal{B}$ too.
The Main Theorem. For a $T_1$-space $X$ to be homeomorphic to a dense subspace of a connected totally ordered space, it is necessary and sufficient that $X$ have a connected linear base.