The authors consider the subgeometries generated by special subsets of a topological generalized quadrangle \(Q\) [as defined by the reviewer and \textit{N. Knarr} in Topology Appl. 34, No. 2, 139-152 (1990; Zbl 0692.51008)]. For example, they show that every nonempty open subset generates \(Q\), if \(Q\) is not discrete. The paper aims at applications to orbits of connected automorphism groups, in the spirit of the work of Salzmann and others on compact connected projective planes [compare \textit{H. Salzmann, D. Betten, Th. Grundhöfer, H. Hähl, R. Löwen}, and \textit{M. Stroppel}, `Compact projective planes,' De Gruyter, Berlin (1996; Zbl 0851.51003)]. In particular, it is proved that each nontrivial connected automorphism group of a compact connected generalized quadrangle \(Q\) has two orbits on points or lines which together generate a weak subquadrangle of \(Q\) of positive topological dimension. In another paper by the authors [Arch. Math. 66, 77-79 (1996)], this is used to prove that the automorphism group of \(Q\) has finite topological dimension, if \(Q\) has finite topological dimension.