On the definition of topological polygons
Let \(Q=(P,L)\) be a generalized \(n\)-gon with vertex set \(V: =P \cup L\). For each \(1\leq k<n\) denote by \(D_k\) the set of all pairs of vertices of \(Q\) at distance \(k\). By the very definition of a generalized \(n\)-gon, for each \((x,y) \in D_k\) there exists a unique path of length \(k\) from \(x\) to \(y\), and we denote the vertices of this path by \(x\), \(f_{k,1} (x,y), \dots, f_{k,k-1} (x,y),y\). Now we equip \(P\) and \(L\) with topologies. Then \(Q\) is defined to be a topological \(n\)-gon if all mappings \(f_{k,i}: D_k \to V\) \((1\leq i<k<n)\) are continuous. The author proves the following useful theorem: If \(f_{n-1,i} :D_{n-1} \to V\) is continuous for each \(1\leq i<n-1\) and if the set \(D_{n-1}\) is open in \(V \times V\), then \(Q\) is a topological \(n\)-gon.
scientific article; zbMATH DE number 913038