Topologies defined using ideals

In this thesis we study the topologies formed by a modification of some given topology using ideals - we focus on localizable and strongly localizable ideals. In the first chapter we use a certain set mapping to define ideal topology, then we show its relation to the initial topology. Next we investigate what properties the elements of ideal obtains in the new topology, for example on certain conditions the ideal becomes exactly the set of all nowhere dense sets in the ideal topology. Finally, we show when the new topology is regular and formulate necessary and sufficient conditions for a set with ideal topology to be a Baire space. In the second chapter we apply the results on concrete examples of ideals and topologies defined by them