Topological model theory with an interior operator: Consistency properties and back — and forth arguments

The logicL(I) is introduced. Compactness and Löwenheim-Skolem theorems for topological models are proved. Several axiomatizations are given and proved to be complete. Interpolation- and preservation theorems are proved using consistency properties. A Back and Forth criterion for elementary equivalence is proved. Most of the results are extended (appropriately) to infinitary logic.