Gated sets in quasi-metric spaces

In 1987, Dress and Scharlau studied the well-known concept of gated sets in metric spaces in order to investigate projections in Tits buildings. In this article, we introduce analogously the concept gated sets in a T 0 -quasi-metric space that we call in-gated set. It turns out that in a T 0 -quasi-metric space, there is a dual concept which we call outgated set. We use these concepts to extend some classical results on gluing a family of hyperconvex metric spaces along a set such that the resulting space preserves hyperconvexity from a metric point of view to quasi-metric settings.