More on rc-Lindelöf sets and almost rc-Lindelöf sets

A subset A of a space X is called regular open if A = IntA, and regular closed if X\A is regular open, or equivalently, if A= IntA. A is called semiopen [16] (resp., preopen [17], semi-preopen [3], b-open [4]) ifA⊂ IntA (resp.,A⊂ IntA,A⊂ IntA ,A⊂ IntA∪ IntA). The concept of a preopen set was introduced in [6] where the term locally dense was used and the concept of a semi-preopen set was introduced in [1] under the name β-open. It was pointed out in [3] that A is semi-preopen if and only if P ⊂ A⊂ P for some preopen set P. Clearly, every open set is both semiopen and preopen, semiopen sets as well as preopen sets are b-open, and b-open sets are semi-preopen. A is called semiclosed (resp., preclosed, semi-preclosed, b-closed) if X\A is semiopen (resp., preopen, semi-preopen, b-open). A is called semiregular [8] if it is both semiopen and semiclosed, or equivalently, if there exists a regular open set U such that U ⊂A⊂U . Clearly, every regular closed (regular open) set is semiregular. The semiclosure (resp., preclosure, semi-preclosure, b-closure) denoted by sclA (resp., pclA, spclA, bclA) is the intersection of all semiclosed (resp., preclosed, semi-preclosed, b-closed) subsets of X containingA, or equivalently, is the smallest semiclosed (resp., preclosed, semi-preclosed, b-closed) set containing A. Dually, the semi-interior (resp., preinterior, semi-preinterior, b-interior) denoted by sint A (resp., pintA, spintA, bintA) is the union of all semiopen (resp., preopen, semi-preopen, b-open) subsets of X contained in A, or equivalently, is the largest semiopen (resp., preopen, semi-preopen, b-open) set contained in A. A function f from a space X into a space Y is called almost open [20] if f −1(U) ⊂ f −1(U) whenever U is open in Y , semicontinuous [16] if the inverse image of each