]{} That $x^{1}$ is incident to $S^{0}$ means that there is a 0-path $P_{x}^{0}$ in $S^{0}$ that either reaches $x^{1}$ through a 0-tip of $x^{1}$ or reaches $x^{1}$ through a branch. Similarly, there is such a 0-path $P_{y}^{0}$ reaching $y^{1}$. Let $u^{0}$ be a node of $P_{x}^{0}$, and let $v^{0}$ be a node of $P_{y}^{0}$. Since $S^{0}$ is 0-connected, there is a 0-path $P_{uv}^{0}$ in $S^{0}$ terminating at $u^{0}$ and $v^{0}$ (possibly a trivial 0-path if $u^{0}=v^{0}$). Then, $P_{x}^{0}\cup P_{uv}^{0}\cup P_{y}^{0}$ as a 0-walk in $S^{0}$ as asserted. $\Box$

A [*nontrivial, two-ended 1-walk*]{} $W^{1}$ is a finite sequence: $$W^{1}\;=\;\langle x_{0},W^{0}_{0},x_{1}^{1},W_{1}^{0},\ldots,x_{m-1}^{1},W_{m-1}^{0},x_{m}\rangle  \label{6.2}$$ with $m\geq 1$ that satisfies the following conditions.

[1.]{} $x_{1}^{1},\ldots,x_{m-1}^{1}$ are 1-nodes, while $x_{0}$ and $x_{m}$ may be either 0-nodes or 1-nodes.

[2.]{} For each $k=0,\ldots,m-1$, $W_{k}^{0}$ is a nontrivial 0-walk that reaches the two nodes adjacent to it in the sequence.

[3.]{} For each $k=1,\ldots,m-1$, at least one of $W_{k-1}^{0}$ and $W_{k}^{0}$ reaches $x_{k}^{1}$ through a 0-tip, not through a branch.

A [*one-ended*]{} 1-walk is a sequence like (\[6.2\]) except that it extends infinitely to the right. An [*endless*]{} 1-walk extends infinitely on both sides. A [*trivial 1-walk*]{} is a singleton set whose sole element is either a 0-node or a 1-node.

We now define a more general kind of connectedness (called “1-wconnectedness” to distinguish it from path-based 1-connectedness). Two branches (resp. two nodes—either 0-nodes or 1-nodes) will be said to be 1-[*wconnected*]{} if there exists a 0-walk or 1-walk that terminates at a 0-node of each branch (resp. that terminates at those two nodes). If a terminal node of a walk is the same as, or contains, or is contained in the terminal node of another walk, the two walks taken together form another walk. We call this the [*conjunction*]{} of the two walks. It follows that 1-wconnectedness is 