Topology idealized

Inspired by recent work of A. Njamcul and A. Pavlović, we study topological spaces expanded by ideals of sets. We investigate the relationship between the cardinal invariants of the ideal topological space (X, τI), the underlying topology $(X, τ)$, and the cardinal characteristics of the ideal I. Furthermore, we analyze continuous maps between standard Polish spaces and their ideal expansions. We establish rigidity phenomena showing that continuous functions from $(X, τ)$ into (X, τI) are severely constrained (e.g., taking at most countably many values on Polish spaces and being strictly constant on R), whereas continuous functions from (R, τI) to (R, τ) collapse to standard continuous functions under mild conditions on the ideal.