Finite Unions of D-Spaces and Applications of Nearly Good Relation

Some results are obtained on finite unions ofD-spaces. It is proved that if a space is the union of finitely many locally compact Dsubspaces, then it is aD-space. It follows that a space is aD-space if it is the union of finitely many locally compact submetacompact subspaces. And a space is a D-space if it is the union of a D-subspace with a locally compact D-subspace. This partially answers one problem raised by Arhangel’skii. At last, some examples are given to exhibit the applications of nearly good relation to discover D-classes.

Among the topics on -spaces, the addition theorems occupy an important role.It has been an interesting subject, especially since Arhangel'skii raised the problem in [2,3] whether the union of two -subspaces is also .In Section 3, we mainly consider the problem in locally compact spaces and give a partial answer by showing that a space is  provided that it is the union of a -subspace with a locally compact -subspace.Besides, we obtain that if a space is the union of finitely many locally compact -subspaces, then it is a -space.With its help, it is shown that a space is a -space if it is the union of finitely many locally compact submetacompact subspaces.
Another important task in studying -spaces is to discover typical -classes.The method is a key to do this work, and hence some methods and related concepts emerged during the process.Among them, we believe that the concept of nearly good relation is an important one, which was introduced in [6] by Gruenhage and helped to obtain that any space satisfying open  is .Unfortunately, the concept did not attract much attention, and hence it is difficult to find other results based on the construction of nearly good relations.In fact, it can help us discover some -classes easily.In Section 4, we exhibit this with some examples.Moreover, we hope that our work will interest others in studying spaces and related open problems.
All spaces are assumed to be  1 -spaces.

Definitions
For the purpose of convenience, we recall the following definitions.In 1965, Worrell Jr. and Wicke in [12] introduced the class of -refinable spaces.Since this class generalizes paracompact spaces, metacompact spaces, and submetacompact spaces, Junnila suggested in [13] that this class be renamed submetacompact spaces; we use this name in the following.Definition 5 (see [7]).A relation  on a space  (resp., from  to [] < ) is nearly good if  ∈  implies  for some  ∈  (resp.,  ỹ for some ỹ ∈ [] < ).Definition 6 (see [6]).Given a neighborhood assignment  on , a subset  of  is -close if ,   ∈  ⇒  ∈ (  ) (equivalently,  ⊂ () for every  ∈ ).Definition 7 (see [6]).A family L of subsets of a space  is point-countably expandable if there exists an open family {  :  ∈ L} such that  ⊂   for each  ∈ L and { ∈ L :  ∈   } for each  ∈ .Definition 8 (see [14]).A topological space (, T) is metrizable if there exists a metrizable topology  on  with  ⊂  and an assignment Definition 9 (see [14]).A cover L of a topological space  is thick if it satisfies the following condition.One can assign L() ∈ L < and ( Remark 10.In a  1 -space, the assignment "L() ∈ L < " in the previous condition can be weakened to "L() Definition 11 (see [15]).A family P = ⋃ ∈ P  of subsets of  is a weak base of , if the following conditions holds.In the following two sections, we denote by  the closure of  in the whole space and by Cl   the closure of a set  in the space .Besides, denote by N the set of all positive natural numbers.
About other terminologies and notations that are omitted here, please refer to [16].

Finite Union of Locally Compact 𝐷-Spaces
Theorem 12.If a space  is the union of finitely many locally compact -subspaces, then it is a -space.
Proof.Suppose that  = ⋃  =1   , where each   is a locally compact -space.
To prove that  is a -space, let  be a neighborhood assignment on .We prove inductively and suppose that  =  1 ∪  2 firstly.
Proof of Claim 1. Denote  =  1 , and take an  ∈ and assume on the contrary that  ∈ , which would follow that every neighborhood intersects  and thus intersects  1 , a contradiction with the fact  ∉  ⊂  1 .
As a closed subspace of the -space  2 , the space  is a -space.Then there exists a closed discrete subset  1 of , such that ∪( 1 ) ⊃ .Moreover, by Claim 2,  is closed in , so the set  1 is also closed and discrete in .
Clearly, the set  =  \ ∪( 1 ∪  2 ) is closed in  and contained in  2 .Then there exists a closed discrete subset  3 in , and thus in , such that ∪( 3 ) ⊃ .
It is trivial to check that  =  1 ∪  2 ∪  3 is closed and discrete in .Moreover, () is a cover of  since  =  ∪  ∪ .Therefore,  is a -space, and we complete the proof for the case  = 2.
For the case  > 2, assume inductively that ⋃ −1 =1   is a -subspace.Since the subspace   is locally compact and open in its closure, with a similar construction as foregoing process, we can obtain a closed and discrete subset  of  such that () covers .And thus  is a -space.
As a corollary of Theorem 12, we have the following consequence.
Corollary 13.Suppose that  = ⋃  =1   , where each   is a submetacompact locally compact subspace.Then  is a space.
Proof.Since compact space is  and   is locally compact for any 1 ≤  ≤ , then every point of   has a neighborhood which is -subset; that is, the space   is locally .Moreover, because every locally submetacompact -space is  [16, Theorem 5.10], each   is a -space.By Theorem 12, as the union of finitely many locally compact -spaces, the space  is a -space.
In fact, we see from the proof of Theorem 12 that, when  =  1 ∪  2 , the result can be obtained even only  1 or  2 is locally compact.So we have the following result, which is a partial answer to the problem whether a space is a -space when the space is the union of two -subspaces [3, Problem 1.1].Theorem 14. Suppose that  =  1 ∪  2 , where  1 and  2 are all -subspace and one of them is locally compact.Then  is a -space.

Applications of Nearly Good Relation in Discovering 𝐷-Classes
The following result presents us a method to discover spaces and we will show its use in this section with some examples.And we hope it will remind others with the use of nearly good relation in the study of -spaces.
It is well known that every space with countable base is a -space.In this section, we mainly show that some general properties can also imply .
Firstly, the following result shows that many spaces with point-countable networks have -property.

Proposition 16. Every space with a point-countably expandable network is a 𝐷-space.
Proof.Assume that  has a point-countably expandable network L and the open family G = {  :  ∈ L} satisfies that  ⊂   for each  ∈ L and { ∈ L :  ∈   } is countable for each  ∈ .
To show that  is a -space, let  be an arbitrary neighborhood assignment on .Define a relation on  as follows: ⇐⇒ ∃ ∈ L, such that  ∈  ⊂  () ,  ∈   . ( To show that  is nearly good, let  ⊂ , and let  ∈ .Since L is a network of , there exists  ∈ L such that  ∈ .Then   is an open neighborhood of , and hence   ∩  ̸ = 0. Then there exists  ∈   ∩ .It follows that  and  are nearly good. For each  ∈ L, let () = { :  ∈  ⊂ ()}.Then for every  ∈ (), we have that () ⊃ (); that is, () is a close set.By the definition of the relation , it is easy to check that  −1 () = ⋃ ∈  () is a countable union of -close set.Hence by Proposition 15, there exists a closed discrete subset  of  such that ⋃ () = .And thus  is a -space.
In [14], a well-behaved class: -metrizable spaces were introduced and then were proved in [17] to have -property.Besides, as another good generalization of point-countable base, the point-countable weak base also implies -property shown in [18].However, the proofs of both results are very complicated.With the help of constructions of nearly good relations, we can prove them much easier.
Proof.Suppose that  is a -metrizable space.Since every subspace of  is -metrizable (see the remark following [14,Theorem 3.4]), we only need to show that  is a -space.
To show that  is a -space, let  be an arbitrary neighborhood assignment on .Define a relation from  to [] < as follows: To show that  is nearly good, let  ⊂ , and let  ∈ .Since F is a network of , there exist  ∈  and   ∈ F  such that  ∈   ⊂ ().
We have that  ⊂ ⋃{  () :  ∈ [] < }, and it follows that there exists  ∈ [] < such that  ∈   () = ⋃ F  ().Moreover, since F  is a partition of  and  ∈   ∈ F  , then   ∈ F  ().Therefore such  and   witness that .We have shown that  is a nearly good relation.
By Proposition 15, there exists a closed discrete subset  of  such that ⋃ () = .We have shown that  is a space.
Proposition 18 (see [18]).Every space with a point-countable weak base is a -space.
Proof.Suppose that  has a weak base P = {P  :  ∈ } such that { ∈ P :  ∈ } is countable for every  ∈ .
We call a finite family {   : 1 ≤  ≤ } a chain of length  from  to  if, for every 1 ≤  ≤ ,  +1 ∈    for some    ∈ P   where we denote  =  1 and  =  +1 .
To show that  is a -space, let  be an arbitrary neighborhood assignment on  and define a relation on  in the following way:  ⇐⇒ ∃ a chain {   : 1 ≤  ≤ } from  to , where  1 = . (5) To show that  is nearly good, let  ⊂  and  ∈ .We construct a neighborhood of  as follows.
Step 2. For every  ∈   taken in Step 1, take a   ∈ P  .
Inductively, we take other sets in P in following steps.Assume that Step  − 1 has been finished, and now we go to Step .
Step n.For every  taken in Step  − 1 and every  ∈ , take a   ∈ P  .
Denote by  the union of the set  taken in all steps.Then  is a open neighborhood of .Indeed, for every  ∈ , there must exist an  ∈ N and  taken at Step  such that  ∈ ; then at Step +1, one   ∈ P  will be taken, and thus   ⊂ .
Since  ∈  and  is an open neighborhood of , then there exists  ∈  such that  ∈ .It follows from the definition of  that .Thereby, the relation  is nearly good.
Since P is point countable, then for each  ∈  and  ∈ N, the set   () = { ∈  : , and the length from  to  is } is countable.It follows that  −1 () = ⋃ ∈N   () is countable, and hence it is the union of countable union of -close set.
(a) For every  ∈ ,  ∈ ∩P  .(b) If ,  ∈ P  , there exists  ∈ P  such that  ⊂  ∩ .(c) A set  is open in  if and only if, for every  ∈ , there exists  ∈ P  such that  ⊂ .
Definition 2. A space is locally compact if each point in  has a compact neighborhood.
Definition 1.A neighborhood assignment is a mapping from a space to its topology.A space is called a -space if, for every neighborhood assignment  on , there exists a closed discrete subset  of  such that the family () = {() :  ∈ } covers .
Definition 4. A sequence ⟨L  ⟩ of covers of a space  is sequence if, for every  ∈ , there exists some  ∈  such that the family L  is point finite at .A space  is called submetacompact, if every open cover of  has a -sequence of open refinement.Note that, for a set , define [] < = { ⊂  : || < }.