Sets with Steinhaus and Smital properties

Density, Smital Property and Quasicontinuity

“…For (5), observe that from DS I (F ) = DS I (F ) and (1) we obtain F ⊂ DS (2) I (F ). Hence, by (2), DS (1) I (F ) ⊃ DS (3) I (F ). But, again from DS (2) I (F ) = DS (2) I (F ) and (1), it follows that DS (1) I (F ) ⊂ DS (3) I (F ).…”

“…Similarly, is it true that DS (2) K (B\K) = (B\K)↑? The negative answer for both questions was supplied recently by Filipczak et al in [3]. Example 5.3.…”