İdeal topological spaces and i̇deal extensions

One way to obtain topology is to determine closed sets on a set with the Kuratowski closure operator. A Kuratowski closure operator can be obtained with the topology on a set, the ideal defined on this set and the local function defined by this ideal. With these local functions obtained, we can produce a finer topology than the initial topology. Using free ideals, a space homeomorphic to a topological space and an extension of this space can be obtained with an ideal extension. With new concepts defined together with free ideals, the compactness of this space can be determined. In this study, the ideal topological space generated by an ideal I, the extension of ideals, and the compactness of the topology defined on the power system are examined.