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 containing A, 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


Introduction and preliminaries
A subset A of a space X is called regular open if A = Int A, and regular closed if X\A is regular open, or equivalently, if A = Int A. A is called semiopen [16] (resp., preopen [17], semi-preopen [3], b-open [4] 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.
A function [16] if the inverse image of each 2 More on rc-Lindelöf sets and almost rc-Lindelöf sets open set is semiopen, β-continuous [1] if the inverse image of each open set is β-open, weakly θ-irresolute [13] if the inverse image of each regular closed set is semiopen, rccontinuous [14] if the inverse image of each regular closed set is regular closed, and wrccontinuous [2] if the inverse image of each regular closed set is semi-preopen.We will use the term semiprecontinuous to indicate β-continuous.Clearly, every semicontinuous function is semi-precontinuous, every rc-continuous function is weakly θ-irresolute, and every weakly θ-irresolute function is wrc-continuous.It is also easy to see that a function that is both semicontinuous (resp., semi-precontinuous) and almost open is weakly θ-irresolute (resp., wrc-continuous).
A function f from a space X into a space Y is called somewhat continuous [12] A space X is called a weak P-space [18] if for each countable family {U n : n ∈ N} of open subsets of X, ∪U n = ∪U n .Clearly, X is a weak P-space if and only if the countable union of regular closed subsets of X is regular closed (closed).
A space X is called rc-Lindelöf [15] (resp., nearly Lindelöf [5]) if every regular closed (resp., regular open) cover of X has a countable subcover, and called almost rc-Lindelöf [10] if every regular closed cover of X has a countable subfamily whose union is dense in X.
A subset A of a space X is called an S-set in X [7] if every cover of A by regular closed subsets of X has a finite subcover, and called an rc-Lindelöf set in X (resp., an almost rc-Lindelöf set in X) [9] if every cover of A by regular closed subsets of X admits a countable subfamily that covers A (resp., the closure of the union of whose members contains A).Obviously, every S-set is an rc-Lindelöf set and every rc-Lindelöf set is an almost rc-Lindelöf set; it is also clear that a subset A of a weak P-space X is rc-Lindelöf in X if and only if it is almost rc-Lindelöf in X.
Throughout this paper, N denotes the set of natural numbers.For the concepts not defined here, we refer the reader to Engelking [11].
In concluding this section, we recall the following facts for their importance in the material of our paper.
Theorem 1.2 [9].Let A be a preopen subset of a space X and B ⊂ A. Then B is rc-Lindelöf (resp., almost rc-Lindelöf) in X if and only if B is rc-Lindelöf (resp., almost rc-Lindelöf) in A. In particular, a preopen subset A of a space X is rc-Lindelöf (resp., almost rc-Lindelöf) in X if and only if A is an rc-Lindelöf (resp., almost rc-Lindelöf) subspace.
Mohammad S. Sarsak 3 Proposition 1.5 [9].A subset A of a space X is rc-Lindelöf (resp., almost rc-Lindelöf) in X if and only if every cover of A by semiopen subsets of X admits a countable subfamily the union of the closures of whose members (resp., the closure of the union of whose members) contains A.
Proposition 1.6 [19].Let A be a preopen, almost rc-Lindelöf set in a space X and B a regular closed subset of X, then A ∩ B is almost rc-Lindelöf in X.In particular, a regular closed subset A of an almost rc-Lindelöf space X is almost rc-Lindelöf in X.

Further properties
This section is devoted to study new properties concerning rc-Lindelöf sets and almost rc-Lindelöf sets.We obtain several characterizations of rc-Lindelöf sets and almost rc-Lindelöf sets.
The following proposition is an improvement of Proposition 1.6 and the fact of Theorem 1.1 that a regular open subset of an almost rc-Lindelöf space X is almost rc-Lindelöf in X.
Proposition 2.1.Let A be a preopen, almost rc-Lindelöf set in a space X and B a semiregular subset of X, then A ∩ B is almost rc-Lindelöf in X.In particular, a semiregular subset A of an almost rc-Lindelöf space X is almost rc-Lindelöf in X.
Proof.Since B is a semiregular subset of X, there exists a regular open subset U of X such that U ⊂ B ⊂ U, thus by Lemma 1.7, it follows that Since A is almost rc-Lindelöf set in X, it follows from Theorem 1.1 that A ∩ U is almost rc-Lindelöf set in X.The result yields from Proposition 1.3.Proposition 2.2 [19].If A is a regular closed subset of a space X such that A is almost rc-Lindelöf in X, then A is an almost rc-Lindelöf.
The following proposition includes an improvement of Proposition 2.2.Proposition 2.3.Let A be a semiopen subset of a space X and B ⊂ A. If B is rc-Lindelöf (resp., almost rc-Lindelöf) in X , then B is rc-Lindelöf (resp., almost rc-Lindelöf) in A. In particular, if A is a semiopen subset of a space X such that A is rc-Lindelöf (resp., almost rc-Lindelöf) in X, then A is an rc-Lindelöf (resp., almost rc-Lindelöf) subspace.
Proof.Follows from Proposition 1.5 and the fact that if A is a semiopen subset of a space X and B is semiopen in A, then B is semiopen in X.
Corollary 2.4 [2].Let X be an rc-Lindelöf weak P-space.If U ⊂ A ⊂ U, where U is a regular open subset of X, then A is an rc-Lindelöf subspace.
Proof.By Theorem 1.1, U is an rc-Lindelöf set in X and thus almost rc-Lindelöf in X.By Proposition 1.3, A is almost rc-Lindelöf in X, but X is a weak P-space, so A is rc-Lindelöf in X.Finally, since A is semiopen (it is moreover semiregular), it follows from Proposition 2.3 that A is an rc-Lindelöf subspace.
4 More on rc-Lindelöf sets and almost rc-Lindelöf sets The following theorem includes new characterizations of rc-Lindelöf sets and almost rc-Lindelöf sets.
Theorem 2.5.Let A be a subset of a space X.Then the following are equivalent.
(ii) Every cover of A by semi-preopen subsets of X admits a countable subfamily the union of the closures of whose members (resp., the closure of the union of whose members) contains A. (iii) Every cover of A by b-open subsets of X admits a countable subfamily the union of the closures of whose members (resp., the closure of the union of whose members) contains A. (iv) Every cover of A by semiopen subsets of X admits a countable subfamily the union of the closures of whose members (resp., the closure of the union of whose members) contains A. (v) Every cover of A by semiregular subsets of X admits a countable subfamily the union of the closures of whose members (resp., the closure of the union of whose members) contains A.
Proof.(i)⇒(ii): follows since the closure of a semi-preopen set is regular closed.
The following theorem also characterizes rc-Lindelöf sets and almost rc-Lindelöf sets, it is a direct consequence of Theorem 2.5 and the definition of rc-Lindelöf (almost rc-Lindelöf) sets.
Theorem 2.6.Let A be a subset of a space X.Then the following are equivalent.
(ii) If U ∼ = {U α : α ∈ Λ} is a family of regular open subsets of X satisfying that for any countable subcollection

Invariance properties
In this section, we mainly study several types of functions that preserve the property of being an rc-Lindelöf (almost rc-Lindelöf) set.Mohammad S. Sarsak 5 Definition 3.1 [19].A function f from a space X into a space Y is said to be slightly In [19], it was shown that if a function f : X → Y is slightly continuous and weakly θ-irresolute, then f (A) is almost rc-Lindelöf in Y whenever A is almost rc-Lindelöf set in X.The following theorem is analogous to this result; it has a similar proof that we will mention for the convenience of the reader.Theorem 3.2.Let f : X → Y be a slightly continuous and weakly θ-irresolute function.It will be seen later that the condition slightly continuous of Corollary 3.4 is not essential for preserving the almost rc-Lindelöf property.
Corollary 3.5 [2].Let f : X → Y be a surjective, continuous, and almost open function.If X is rc-Lindelöf, then Y is rc-Lindelöf.
Obviously, every continuous function is both semicontinuous and slightly continuous.However, the converse is not true as the following example tells.
Proof.Let U be open in X.Then scl(U) = U ∪ intU = U (as X is extremally disconnected).Since f is semicontinuous, it follows that f (scl(U)) = f (U) ⊂ f (U).Hence f is slightly continuous.
The following corollary is an immediate consequence of Corollary 3.4 and Proposition 3.7.
Corollary 3.8 [2].Let f : X → Y be a semicontinuous, almost open surjection, where X is extremally disconnected.If X is rc-Lindelöf, then Y is rc-Lindelöf.