Abstract: Fibres of a morphism between complex spaces form a family that encodes much information regarding the behaviour of the morphism. For example, knowing only about the variation of the topological dimension of the fibres suffices to determine whether a mapping is open or not (Remmert Open Mapping Theorem). This has lead us to an efficient algebraic method of testing for openness by means of the blow-up mapping, and successively, to a very efficient method of testing for flatness (joint work with Janusz Adamus). Apart from merely detecting non-openness, I am also trying to study different modes of being non-open for an analytic map, especially in the general setting of maps over a singular target.