MAPPINGS AND DECOMPOSITIONS OF CONTINUITY ON ALMOST LINDELÖF SPACES

Among the various covering properties of topological spaces a lot of attention has been given to those covers which involve open and regularly open sets. In 1982 Balasubramanian [4] introduced and studied the notion of nearly Lindelöf spaces and in 1984 Willard and Dissanayake [21] gave the notion of almost Lindelöf spaces. Then in 1996 Cammaroto and Santoro [5] studied and gave further new results about these spaces which are considered as one of the main generalizations of Lindelöf spaces. Moreover, decompositions of continuity have been recently of major interest among general topologists. They are being studied by many authors, including Singal and Singal [19], Popa and Stan [16], Noiri [14], Long and Herrington [11], Mashhour et al. [12], Abd El-Monsef et al. [1], Dontchev [6], Dontchev and Przemski [7], Nasef and Noiri [13], Park and Ha [15], and Baker [2, 3]. In fact, mathematicians introduced in several papers different and interesting new decompositions of continuity as well as generalized continuous functions. Throughout the present paper, spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated. The topological space (X ,τ) will be replaced by X if there is no chance for confusion. The interior and the closure of any subset A of (X ,τ) will be denoted by Int(A) and Cl(A), respectively. The purpose of this paper is to study the effect of mappings and some decompositions of continuity on almost Lindelöf spaces. We also show that some mappings preserve this property. The main result is that the image of an almost Lindelöf space under a θ-continuous function is almost Lindelöf.


Introduction
Among the various covering properties of topological spaces a lot of attention has been given to those covers which involve open and regularly open sets.In 1982 Balasubramanian [4] introduced and studied the notion of nearly Lindelöf spaces and in 1984 Willard and Dissanayake [21] gave the notion of almost Lindelöf spaces.Then in 1996 Cammaroto and Santoro [5] studied and gave further new results about these spaces which are considered as one of the main generalizations of Lindelöf spaces.
Throughout the present paper, spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated.The topological space (X,τ) will be replaced by X if there is no chance for confusion.The interior and the closure of any subset A of (X,τ) will be denoted by Int(A) and Cl(A), respectively.
The purpose of this paper is to study the effect of mappings and some decompositions of continuity on almost Lindelöf spaces.We also show that some mappings preserve this property.The main result is that the image of an almost Lindelöf space under a θ-continuous function is almost Lindelöf.

Preliminaries
Recall that a subset A ⊆ X is called regularly open (regularly closed) if A = Int(Cl(A)) (A = Cl(Int(A))).The topology generated by the regularly open subsets of a space (X,τ) is called the semiregularization of (X,τ) and is denoted by (X,τ s ).A space (X,τ) is said to be semiregular if the regularly open sets form a base for the topology or equivalently τ = τ s .It is called almost regular if for any regularly closed set C and any singleton {x} disjoint from C, there exist two disjoint open sets U and V such that C ⊆ U and x ∈ V .Note that a space X is regular if and only if it is semiregular and almost regular [17].Moreover, a space X is said to be submaximal if every dense subset of X is open in X and it is called extremally disconnected if the closure of each open set of X is open in X.A space X is said to be mildly normal [20] if whenever A and B are disjoint regularly closed sets in X, then there are two disjoint open sets U and V with A ⊆ U and B ⊆ V .And it is called nearly paracompact [18] if every regularly open cover of X admits an open locally finite refinement.Definition 2.1 (see [4]).A topological space X is said to be nearly Lindelöf if for every open cover {U α : α ∈ Δ} of X, there exists a countable subset {α n : n ∈ N} ⊆ Δ such that X = n∈N Int(Cl(U αn )).That is, every regularly open cover of X admits a countable subcover.
It illustrates the relations among some of these mappings: Moreover, it is clear that every strong θ-continuous mapping is θ-continuous and δcontinuous.It is clear also, that every δ-continuous mapping is almost δ-continuous but the converse is not true as Example 3.5 below shows.Contra-continuity implies subconracontinuity but the converse, in general, is not true (see [3]).So, with Lemma 3.12 and Example 3.13 below, we obtain the following diagram in which none of these implications is reversible:

Almost Lindelöf spaces
Definition 3.1 (see [21]).A topological space X is said to be almost Lindelöf if for every open cover {U α : α ∈ Δ} of X there exists a countable subset Note that every Lindelöf space is nearly Lindelöf and every nearly Lindelöf space is almost Lindelöf but the converses, in general, are not true (see [5]).Moreover, it is well known that the continuous image of a Lindelöf space is Lindelöf and in [8] it was shown that the δ-continuous image of a nearly Lindelöf space is nearly Lindelöf.In the case of almost Lindelöf spaces we give the following theorem.
This implies that Y is almost Lindelöf and completes the proof.
Corollary 3.3.The θ-continuous image of an almost Lindelöf space is almost Lindelöf.
The converse of Lemma 3.4 is not true, in general, as the following example shows.
Corollary 3.8.Let f : X → Y be θ-continuous surjection from X into an almost regular space Y .If X is almost Lindelöf, then Y is nearly Lindelöf, mildly normal, and nearly paracompact.
is an open cover of the almost Lindelöf space X.So there exists a countable subset {U αx n : n ∈ N} such that This implies that Y is Lindelöf and completes the proof.
Corollary 3.10.The strong θ-continuous image of an almost Lindelöf space is Lindelöf.
Note that if f : (X,τ) → (Y ,σ) is a strong θ-continuous function, then f is continuous but the converse is not true, in general, as in Example 3.5.
Next we prove that almost δ-continuous image of a nearly Lindelöf space is almost Lindelöf.
) is regularly open in (Y ,σ) containing f (x), then for every x ∈ X, there exists a regularly open subset U αx of X containing x such that f (U αx ) ⊆ Cl(V αx ).So {U αx : x ∈ X} is a regularly open cover of the nearly Lindelöf space X.Thus there exists a countable subset This shows that Y is almost Lindelöf.Now we prove that θ-continuity implies almost δ-continuity.
This implies that f is almost δ-continuous and completes the proof.
The converse of Lemma 3.12 is not true, in general, as the following example shows.
Next we prove the following proposition.
Proposition 3.14.Let f : X → Y be a weakly quasicontinuous and precontinuous surjec- a regularly open cover of the nearly Lindelöf space X.It follows that there exists a countable subset This implies that Y is almost Lindelöf.
We also prove the following proposition.
Proposition 3.15.The image of an almost Lindelöf space under a precontinuous and subcontra-continuous mapping is Lindelöf.
Proof.Let f : X → Y be a subcontra-continuous and precontinuous function from X to Y .Assume that X is almost Lindelöf.Let Ꮾ be an open base for the topology on Y for which f ) is clopen and hence { f −1 (V αx ) : x ∈ X} is a clopen cover of the almost Lindelöf space X.So there exists a countable subfamily {x n : n ∈ N} for which Thus we have This implies that f (X) is Lindelöf and completes the proof.
Since contra-continuous functions are subcontra-continuous, we have the following corollary.
Corollary 3.16.The image of an almost Lindelöf space under a contra-continuous and precontinuous mapping is Lindelöf.
The following two propositions are analogous results of nearly Lindelöf spaces (see [8]).
τ) whereas {b} is regularly open in (X,σ).Since f is θ-continuous, by Lemma 3.12 below, f is almost δ-continuous but it is not δ-continuous since, there is a regularly open set {b} of Y containing f ({a}) but there is no regularly open set U of X containing {a} such that f (U) ⊆ {b}.Corollary 3.6.Let f : X → Y be an almost continuous surjection from X into Y .If X is almost Lindelöf, then Y is almost Lindelöf.