ON GENERALIZATIONS OF REGULAR-LINDELÖF SPACES

We study nearly regular-Lindelöf, almost regular-Lindelöf and weakly regularLindelöf spaces. Characterizations and some properties for these spaces are proposed. Relations among them are also studied. 2000 Mathematics Subject Classification. 54A05, 54D15, 54D20, 54G05.

Note that a nearly Lindelöf space is almost Lindelöf and an almost Lindelöf space is weakly Lindelöf, but none of the two converses is true.Also it is clear that separable spaces are weakly Lindelöf.Moreover, a semiregular and nearly Lindelöf space is Lindelöf [3].Definition 1.2 (see [2]).An open cover {U α : α ∈ ∆} of a topological space X is called regular cover if, for every α ∈ ∆, there exists a nonempty regularly closed subset C α of X such that C α ⊆ U α and X = α∈∆ Int(C α ).
Obviously, every almost Lindelöf space is almost regular-Lindelöf, but the converse is not true, in general (see [3,Example 3.11 and Property 4.5]).Also weakly Lindelöf spaces need not be almost regular-Lindelöf (see Example 4.5 below).

Remark 2.2. Every regular cover {U
a regularly open cover of the almost regular space X and U αx containing x, there exist two regularly open subsets V αx and W αx such that x ∈ V αx ⊆ Cl(V αx ) ⊆ W αx and Cl(W αx ) ⊆ U αx (see [5,Theorem 2.2]).So the family {W αx : x ∈ X} is a regular cover of X that refines {U α : α ∈ ∆}.Proposition 2.4 (see [3]).An almost regular and almost regular-Lindelöf space X is nearly Lindelöf.Corollary 2.5.A regular and almost regular-Lindelöf space X is Lindelöf.Proposition 2.6.Let (X, τ) be an extremally disconnected and almost regular-Lindelöf space then it is nearly Lindelöf.
Proof.Let {U α : α ∈ ∆} be an open cover of X.Since (X, τ) is extremally disconnected, {Cl(U α ) : α ∈ ∆} is a regular cover of the almost regular-Lindelöf space X.So there exists a countable subset {α n : n ∈ N} ⊆ ∆ such that X = n∈N Cl(U αn ) = n∈N Int(Cl(U αn )).This implies that X is nearly Lindelöf and completes the proof.

Nearly regular-Lindelöf spaces
Definition 3.1 (see [3]).A topological space X is called nearly regular-Lindelöf if every regular cover {U α : α ∈ ∆} of X admits a countable subfamily It is clear that every nearly Lindelöf space is nearly regular-Lindelöf but the converse is not true in general.The space X = (R,σ ) in [3,Example 3.11] was shown not to be almost Lindelöf so it is not nearly Lindelöf.By noting that Int τu (Cl σ (U αn )) ⊆ Int σ (Cl σ (U αn )) where τ u denotes the usual topology on R, we conclude that X = n∈N Int σ (Cl σ (U αn )), which implies that X is nearly regular-Lindelöf.Proposition 3.2.If the space X is almost Lindelöf then it is nearly regular-Lindelöf.
Proof.Let {U α : α ∈ ∆} be a regular cover of X.By Remark 2.2, there exists a family of regularly closed sets open cover of the almost Lindelöf space X that refines {U α : α ∈ ∆}.So there exists a countable subset This implies that X is nearly regular-Lindelöf and completes the proof.Moreover, it is clear that every nearly regular-Lindelöf space is almost regular-Lindelöf.Weakly Lindelöf spaces need not be nearly regular-Lindelöf as Example 4.5 shows.Now using Proposition 2.4, it is easy to prove the following proposition.
Proposition 3.3.An almost regular and nearly regular-Lindelöf space X is nearly Lindelöf.
The following proposition gives a characterization of nearly regular-Lindelöf spaces.

Weakly regular Lindelöf spaces
Definition 4.1 (see [3]).A topological space X is said to be weakly regular-Lindelöf if every regular cover {U α : α ∈ ∆} of X admits a countable subset It is obvious that every weakly Lindelöf space is weakly regular-Lindelöf but for the converse we introduce the following question.Is it true that every weakly regular-Lindelöf space is weakly Lindelöf?We can answer this question with some restrictions on the space as the following proposition shows.Proposition 4.2.A regular and weakly regular-Lindelöf space X is weakly Lindelöf.
Proof.Let {U α : α ∈ ∆} be a regularly open cover of X.For each x ∈ X there exists α x ∈ ∆ such that x ∈ U αx .Since X is almost regular, then by Remark 2.3, there exists a family {W αx : x ∈ X} of regularly open sets in X that forms a regular cover of the weakly regular-Lindelöf space X.So there exists a countable set of points {x 1 ,x 2 ,...,x n ,...} of X such that X = Cl n∈N W αx n ⊆ Cl n∈N U αx n .Since X is semiregular, by [3, Proposition 3.4], X is weakly Lindelöf.Corollary 4.3.Let (X, τ) be a regular space.Then (X, τ) is weakly Lindelöf if and only if (X, τ) is weakly regular-Lindelöf.Proposition 4.4.If (X, τ) is regular and nearly paracompact, then (X, τ) is Lindelöf if and only if (X, τ) is weakly regular-Lindelöf.
The conditions that (X, τ) is regular and nearly paracompact in Proposition 4.4 are necessary.For nearly paracompactness condition, the space E × E of Example 4.5 is regular and weakly regular-Lindelöf but it is not Lindelöf nor nearly paracompact, it is not even almost regular-Lindelöf.For the necessity of the regular condition consider the half-disc topology (see [7,Example 78]).It is known that topology is separable so weakly Lindelöf hence weakly regular-Lindelöf.Also it is neither Lindelöf nor regular.Since the semiregularization of this space is the open upper half-plane with the Euclidean topology which is paracompact, then we conclude that the half-disc topology is nearly paracompact.
It is clear that every almost regular-Lindelöf space is weakly regular-Lindelöf but the converse is not true in general, as the following example shows.
Example 4.5.Let E be the Sorgenfrey line and let E × E be its product with itself which is called the Sorgenfrey plane.It is known that E is regular and separable, so E × E is regular and separable, then it is weakly Lindelöf so weakly regular-Lindelöf.Also it is known that E ×E is regular but not Lindelöf, so by Corollary 2.5, E ×E is not almost regular-Lindelöf, thus it is not nearly regular-Lindelöf.
The following proposition gives a characterization of weakly regular-Lindelöf spaces and its proof is similar to the proof of an analogous result for almost regular-Lindelöf spaces given by Cammaroto and Santoro (see [3,Theorem 4.10]).Proposition 4.6.A topological space X is weakly regular-Lindelöf if and only if, for every family {C α : α ∈ ∆} of closed subsets of X such that, for each α ∈ ∆, there exists an open set A α ⊇ C α with α∈∆ Cl(A α ) = ∅, there exists a countable subfamily {α n : n ∈ N} such that Int( n∈N C αn ) = ∅.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Proposition 3 . 4 .
A space X is nearly regular-Lindelöf if and only if for every family {C α : α ∈ ∆} of regularly closed subsets of X such that, for each α ∈ ∆, there exists an open set A α ⊇ C α with α∈∆ Cl(A α ) = ∅, there exists a countable subfamily {α n : n ∈ N} ⊆ ∆ such that n∈N C αn = ∅.Proof.The proof of Proposition 3.4 is similar to the proof of an analogous result for almost regular-Lindelöf spaces given by Cammaroto and Santoro (see[3,.10]).