Topological sum and some results

This study includes free union of a disjoint non-empty family of topological spaces and research on the discrete union topology.After general information of topology were given and topological spaces were introduced, the release of two comprehensive approaches to free union are discussed. Firstly, subspaces of a given topological space are considered, and topological spaces taken in case of certain conditions are provided by these subspaces are described by the free union. Secondly, it has been told that for a family of disjoint non-empty topological space, the topological space with the discrete topology that will be discussed future, will be free union of the given family.The continuation of the study, research was conducted by taking the second directional approach mentioned above. Primarily, the special case of disjoint union topology has been alluded, then open and closed subsets of free union have been defined, and inner and closing conceptions have been discussed. The part of the research in progress, the results related with base on the topological sum, continuous transformations, separation axioms and convergence are provided with their proofs.With this research, definitions, theorems and some results for topological sum have been obtained by using the known definitions and theorems for the topological spaces.