On the definition of topological polygons
type: Item
instance of: scholarly article
title: On the definition of topological polygons
zbMATH Open document ID: 0857.51004
DOI: 10.1007/BF01222686
published in: Journal of Geometry
publication date: 1996-08-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 913038
zbMATH Keywords: continuity criterion
zbMATH Keywords: compact polygons
zbMATH Keywords: topological \(n\)-gon
author: Rainer Löwen
MaRDI profile type: Publication
cites work: Topology in generalized quadrangles
cites work: Q5201840
cites work: The Nonexistence of Certain Topological Polygons
cites work: Q4855470
cites work: Q4303227
full work available at URL: https://doi.org/10.1007/bf01222686
OpenAlex ID: W1975333965
description: scientific article; zbMATH DE number 913038