Homotopy Properties And Applications Of Mapping In Category Related To GK-topology

Based on the GK-topology on the digital space Z3,i.e.the product topology of two-dimensional grid point topology(GP2-topology)and Khalimsky line topology(K-topology),this thesis makes theoretical analysis and research on 3D digital images.Firstly,under GK-topology,GK-continuous mapping and GK-A-mapping(that is,the connectedness-preserving mapping)are considered,and GK-topology category and GK-adjacency category are introduced.Secondly,GK-homotopy is defined in GK-topology category and the equivalent dis-cription of GK-homotopy is given.It is proved that GK-homotopy is an equivalence relation.Then the GK-homotopy equivalence and GK-contractibility of images are de-fined and the GK-contractibility and homotopy properties of simple GK-path and simple closed GK-curve are discussed.Then,GK-A-homotopy,GK-A-homotopy equivalence,and GK-A-contractibility are defined in GK-adjacent category.The similarities and dif-ferences between the GK-homotopy and GK-A-homotopy are analyzed through specific examples.It is proved that GK-homotopy must be GK-A-homotopy,but not necessarily vice versa.In addition,the homotopy thinning of digital images,especially the digital image contractibility,is considered in GK-adjacency category,and the contractibility or local contractibility of some specific digital images are discussed.Finally,considering the fixed point of mapping in GK-topological category,it is concluded that not every GK-connected digital space X with cardinality greater than or equal to 2 has fixed point property.Considering the fixed point of mapping in GK-adjacent category,it is proved that any GK-A-connected digital space X with cardinality greater than or equal to 2 has no fixed point property.So the concept of almost fixed point property is introduced into GK-adjacent category,and almost fixed point property of some special digital images is considered.