for generalized affine buildings did rely on metric properties as well.

The purpose of the present section is to provide a definition of parallelism for Weyl simplices that does not involve the metric structure of the affine building and which allows a new, combinatorial proof for the existence of a spherical building at infinity. To be precise, in comparison to [@Bennett], we avoid using axiom (A5) in the proof.

\[Def\_buildingAtInfinity\]\[Def\_parallelSimplices\] Let $(X,{\mathcal{A}})$ be a pair satisfying axioms (A1)-(A4). We say that $S$ and $T$ are *parallel* if $S\cap T$ contains a Weyl chamber. We denote by $\partial S$ the parallel class of $S$.

As we will see later on in this section, the set $$\{\partial S : S \text{ Weyl chamber of } X \text{ contained in an apartment of } {\mathcal{A}}\}$$ of equivalence classes of Weyl chambers is the collection of chambers of a spherical building *at infinity of $X$*.

Bennett defined two Weyl simplices to be parallel if they are at bounded Hausdorff distance. One can proof, compare [@Diss 4.23] and [@Bennett], that “being at bounded distance” can be characterized differently.

\[Prop\_parallel\] Given two Weyl chambers $S$ and $T$ the following are equivalent

1.  They are parallel in 