Soft ideal topological spaces

The article deals with the correspondence between soft ideal topological spaces and ideal topological ones. Investigation of soft ideal topological spaces is based on methods of general topology, and the application of results for soft omega open and strongly soft omega open sets is given.


Introduction
initiated the concept of soft sets as a completely different approach for dealing with uncertainties, and over the past few years, the fundamentals of the soft set theory have been studied by many authors. Since the concept of soft topology was introduced by Shabir and Naz (2011), many terms of general topology have found their analogy in soft topological spaces.
There are several papers that document certain problems relating to the fundamentals of the soft set theory and soft topological spaces. Pang 2014, 2015;Shi and Fan 2019) demonstrate the redundancies concerning the increasing popular soft set approaches to general topology, and they claim that soft topology is exactly a special subcase of general topology. Matejdes (2016) also states that a soft topology is nothing more than a topology on Cartesian product, and each soft topological concept has its topological equivalent. Some soft terms (for example, soft homogeneity, soft compactness, soft paracompactness, soft Lindelöfness, soft normality, soft connectedness, soft hyperconnectedness, soft topological sum; see Al Ghour and Bin-Saadon (2019); Al-shami et al. (2020); El-Shafei et al. (2018); Terepeta (2018)) correspond to known, commonly used and studied topological terms. Others (for example, soft separation axioms, see El-Shafei et al. (2018)) correspond to topological terms that bring new challenges to research. In principle,  Alcantud (2021); Matejdes (2021a); Matejdes (2021b); Matejdes (2021c) also document that soft topology is basically part of general topology, and concepts of soft topology can be reduced to the corresponding concepts in topology. More precisely, a soft topological space stemming from a topological space and vice versa is investigated by Alcantud (2021) (with application to base and separability), soft homogeneity is investigated by Matejdes (2021b), enriched soft topology, topological sum, soft regularity, soft compactness and soft Lindelöfness are studied by Matejdes (2021a), and soft continuity is studied in Matejdes (2021c). In all cases, topological counterparts were used as a substitute for soft methods. Based on these facts, we can say that the cumbersome methods used in the theory of soft topological spaces, which are often unnecessary imitations of topological methods, can be effectively overcome by identifying a set-valued mapping with its graph. From this point of view, it is a fundamental transformation of the existing methods used in soft topological spaces into corresponding topological methods. This is a major step forward in improving and simplifying the existing methods used in the current literature.
The aim of the article is to continue the above-mentioned transformations of topological methods into soft topological ones. Namely, the results of ideal topological spaces will be used in the field of soft ideal topological spaces. We especially focus on the results from Al Ghour and Hamed Worood (2020). It should be noted that many other results concerning soft ideal topological spaces (see, for example, Gharib and Abd El-latif (2019); Kandil et al. (2014) where one will find further references to the issue of soft ideal topological spaces) can be investigated within the framework of ideal topological spaces whose known results can be directly applied.
This research article is organized as follows: Sects. 2 and 3 are devoted to the basic concepts of the theory of ideal topological spaces. In order to achieve the specific objectives of the article by Al Ghour and Hamed Worood (2020), we only focus on some results of ideal topological spaces. One may recall these are known facts, but for the sake of completeness and purpose, we present them with proofs unless they are trivial. In Sects. 2 and 3, the choice of ideal topological terms, theorems and lemmas is determined by their use in Sects. 6 and 7 concerning soft omega open and strong soft omega open sets, where the results from Al Ghour and Hamed Worood (2020) are proved as direct consequences.
In Sects. 4 and 5, we show that soft topologies can be fully characterized in terms of standard topologies on a crisp set. This characterization is based on two constructions (see Definition 6, Theorem 4). The first one yields a soft topological space that is associated with any crisp topology on a certain Cartesian product of two sets. The second works in the opposite direction: With any soft topological space, it produces a crisp topology on the Cartesian product.
Both constructions are explicit and ensure a transition from one setting to the other. Their fundamental properties and mutual links are investigated in Theorem 5. We show that such notions as a soft subspace, a soft topological sum, a soft ideal topological space, a soft base and their properties can be transferred from crisp topologies to soft topologies or the other way round (Definition 7, Lemma 7, Lemma 8). This achievement has several remarkable consequences. Concepts from soft topology can be reduced to the corresponding concepts in topology, and results from topological spaces may be exported to soft topological spaces. We give examples of these fundamental advances, namely: soft ω-open sets, strong soft ω-open sets is identified with countable sets, sets with countable sections, respectively (Definition 8, Remark 5).
In Sect. 6, the results are specified into the soft ideals I s and I 0 , and the last section summarizes the results from Al Ghour and Hamed Worood (2020), which are the direct consequence of the obtained results. One may recall that all the results of Al Ghour and Hamed Worood (2020) can be transformed into corresponding topological results and they can be extended for arbitrary soft ideal.

Ideal topological spaces
(X , τ ) denotes a topological space, cl τ (S), int τ (S) the closure (the interior) of S ⊂ X , respectively. If A ⊂ X , then by (A, τ A ) we denote a topological subspace of (X , τ ) where τ A is a subspace topology.
An ideal I on X is a nonempty collection of subsets of X which satisfies the following properties: If A ∈ I and B ⊂ A, then B ∈ I, and if A ∈ I and B ∈ I, then A∪B ∈ I. An ideal topological space is a topological space (X , τ ) with an ideal I on X , and it is denoted by (X , τ, I), see, for example, Al-Omari and Noiri (2013); Ekici and Noiri (2008); Kaniewski et al. (1998), where one can find rich references.
If (X , τ, I) is an ideal topological space and S ⊂ X , then the set of all points in which S is locally not in I with respect to τ , i.e., {x ∈ X : S ∩ U / ∈ I for every open set U containing x} is called the local function of S with respect to τ and I, and it is denoted by D τ,I (S) (denoted also S * (I, τ ), see, for example, Al-Omari and Noiri (2013); Ekici and Noiri (2008); Kaniewski et al. (1998)). Obviously It is clear I A is an ideal on A and I A ⊂ I. An ideal I is called τ -codense, see Kaniewski et al. (1998) Kaniewski et al. (1998).
A topological space (X , τ ) is Lindelöf (weakly Lindelöf, see Frolík (1959)) if every open cover U of X has a countable subfamily V such that X = ∪V (X = cl τ (∪V)). Note U can be replaced by a cover from a base of (X , τ ).
we denote a topology on X , A generated by the base B τ,I , B τ A ,I A , respectively. In the literature, τ I is usually denoted by τ * (I) or briefly τ * . By co X ,I , we denote a family {X \ I : I ∈ I} ∪ {∅}, which is a topology on X .
Remark 1 Let I and J be the ideals on X . Then (1) τ ⊂ B τ,I ⊂ τ I , co X ,I ⊂ B τ,I . Moreover, (X , co X ,I ), (X , τ I ) is a topological space in which any set I ∈ I is closed, respectively (see Lemma 1). It is clear if τ id is the indiscrete topology on X , then (τ id ) I = co X ,I .

Ideals and topologies on the Cartesian product, topological sum
In some cases, we use the notation A Remark 2 In this remark, we specify the families A and B. Let τ be a topology on U , U [e], E × U , respectively, and J , I be an ideal on U , E × U , respectively. Then for any e ∈ E.
Note that ⊕ e∈E σ e is a topology defined as the finest topology on The following lemma will be useful for further investigation, see Engelking (1977).  (Lindelöf, weakly Lindelöf) if and only if E is finite (E is countable) and (U , σ e ) is compact (Lindelöf, weakly Lindelöf) for any e ∈ E.  (5) , S e is locally not in I, , σ e , respectively.
Proof (1): By Remark 2 item (3), item (5) (2): If H ∈ (τ I ) e , then H = G e for some G ∈ τ I . By (1), Theorem 2 Let {(U , σ e ) : e ∈ E} be an indexed family of topological spaces and S ⊂ E × U and I be an ideal on E × U . Then Let Theorem 3 Let {(U , σ e ) : e ∈ E} be an indexed family of topological spaces and I be an ideal on E × U . Then Proof G ∈ (⊕ e∈E σ e ) I if and only if G = ∪ t∈T (G t \ I t ) and G t ∈ ⊕ e∈E σ e , I t ∈ I if and only if (by Lemma 5 item (2) for any e ∈ E if and only if G ∈ ⊕ e∈E (σ e ) I e , by Lemma 5 item (2).

Relations and set-valued mappings
Any subset S of the Cartesian product E × U is a binary relation from a set E to a set U . By R(E, U ), we denote the set of all binary relations from E to U . Two relations A, B are equal if and only if A e = B e for any e ∈ E. The operations of the sum S ∪ T , ∪ t∈T S t , intersection S ∩ T , ∩ t∈T S t , complement S c and the difference S \ T of relations are defined in the obvious way as in the set theory.
By F : E → 2 U we denote a set-valued mapping (multifunction) from E to power set 2 U of U . The set of all set-valued mappings from E to 2 U is denoted by F(E, U ). A set-valued mapping F for which F(e) = {u} and it is empty-valued otherwise is denoted by F u e . If F, G are two set-valued mappings, then F ⊂ G The difference F \ G of F and G is defined as a set-valued mapping given by (F \ G)(e) = F(e) \ G(e) for any e ∈ E. The intersection (union) of family {G t : t ∈ T } of set-valued mappings is defined as a set-valued mapping H : E → 2 U for which H (e) = ∩ t∈T G t (e) (H (e) = ∪ t∈T G t (e)) for any e ∈ E, and it is denoted by ∩ t∈T G t (∪ t∈T G t ). For the intersection (union) of two set-valued mappings F and G, we use On the other hand, any relation S ∈ R(E, U ) determines a set-valued mapping F S from E to 2 U where From the definitions of R G and F S and from the equality of two relations and the equality of two multifunctions, we have It is useful to note the next conditions are equivalent:

Soft ideal topological space and ideal topological space
Definition 3 (Maji et al. 2003;Shabir and Naz 2011) Let E, U be two nonempty sets.
(1) If F : E → 2 U is a set-valued mapping, then F is called a soft set over U with respect to E. A soft set F for which F(e) = ∅ (F(e) = U ) for any e ∈ E is called the null soft set (the full soft set) and F u e is called a soft point. The following theorem deals with the mutual correspondence between ideal topological spaces and soft ideal topological spaces, and plays an important role in the transformation of soft topological problems into topological ones. For the correspondence between topological spaces and soft topological spaces, see Matejdes (2016) ; Matejdes (2021a), Matejdes (2021c).
Theorem 4 There is a one-to-one correspondence between the family of all soft ideal topological spaces over U with respect to E and the family of all ideal topological spaces on E × U as follows:

τ, I) and
(E ×U , R τ , R I )) are mutually corresponding. Similarly, we say a topology τ (a soft topology τ ) and a soft topology F τ (a topology R τ ) (an ideal I (a soft ideal I) and a soft ideal F I (an ideal R I )) are mutually corresponding. If (E ×U , τ 1 , I 1 ) is an ideal topological space and (E, U , τ 2 , I 2 ) is a soft topological space, then they are mutually corresponding if F τ 1 = τ 2 and F I 1 = I 2 if and only if R τ 2 = τ 1 and R I 2 = I 1 .
Any subset of E × U uniquely corresponds to a soft set. The set E × U (∅) corresponds to the full soft set F E×U (the null soft set F ∅ ). Any set from a soft topology τ (a topology τ ) corresponds to an open set (a soft open set) from R τ (F τ ), and its complement corresponds to a closed set (a soft closed set).
The next theorem summarizes the properties of the operators F : R(E, U ) → F(E, U ) and R : F(E, U ) → R(E, U ). For item (1), see the conditions at the end of the previous section and item (2) is trivial. For items (3)-(9), see Matejdes (2021c).
The methods of constructing new topological spaces from old ones and the one-to-one correspondence between the family of topological spaces and soft topological spaces allow the introduction of soft topological spaces. Some of them are introduced in the following definition.

Definition 7
In this definition, we introduce a soft topological sum, a soft topological subspace, and a soft topology corresponding to τ I .
(1) Let {(U , σ e ) : e ∈ E} be an indexed family of topological spaces. A soft topological sum of {(U , σ e ) : e ∈ E} is defined as a soft topology F ⊕ e∈E σ e and it is denoted by ⊕ s e∈E σ e . Note F ⊕ e∈E σ e is equal to {H : E → 2 U : H (e) ∈ σ e for all e ∈ E} which is a soft topology and F σ e [e] = {H : E → 2 U : H (e) ∈ σ e and H ( f ) = ∅ f or f = e} is its soft subbase. So F ⊕ e∈E σ e = ⊕ s e∈E σ e (see notation ⊕ e∈E σ e in Al Ghour and Hamed Worood (2020)). Specially if σ e = J for any e ∈ E, then F ⊕ e∈E J = ⊕ s e∈E J = τ (J) where τ (J) = {F ∈ SS(E, U ) : F(e) ∈ J for any e ∈ E} is a soft topology from Al Ghour and Hamed Worood (2020).
(2) If Y ⊂ U , then a soft topological subspace of (E, U , τ ) on Y is defined as the corresponding soft topological space to a topological subspace (E × Y , (R τ ) E×Y ) where (R τ ) E×Y is a subspace topology on E × Y derived from R τ . A soft topological subspace of (E, U , τ ) on Y is denoted by Al Ghour and Hamed Worood (2020) by (E, Y , τ Y ), see Lemma 9. (3) If (E × U , τ, I) is an ideal topological space, then we can define a soft ideal topological space by (E, U , F τ I ), see Lemma 6 and Lemma 8 (1).

Lemma 6
Let (E × U , τ, I) be an ideal topological space.
In the following, τ ,τ (I,Î) denotes a topology on E × X , a soft topology over U with respect to E (an ideal on E × X , a soft ideal on U with respect to E), respectively. (2) follows from Remark 4.

Soft !-open sets and strongly soft !-open sets
In the next remark, we specify the above results to those of Al Ghour and Hamed Worood (2020)  Definition 8 By I s , I 0 , we denote an ideal of all countable subsets of E × U , an ideal of all subsets I of E × U such that I e ⊂ U is countable for any e ∈ E, respectively. LetÎ 0 ,Î s be the corresponding soft ideal to I 0 , I s , i.e, F I 0 =Î 0 ⇔ RÎ 0 = I 0 , F I s =Î s ⇔ RÎ s = I s , respectively.
Let (E, U ,τ ) be a soft topological space and Y ⊂ U . If F ∈ SS(E, U ), then a soft set F Y is defined as F Y (e) = F(e) ∩ Y for any e ∈ E. A familyτ Y = {F Y : F ∈τ } is called a relative soft topology on Y , see Al Ghour and Hamed Worood (2020). Similarly, we define a soft idealÎ Y = {I Y : I ∈Î} whereÎ is a soft ideal.

Corollary 8
Recall a subset A of (X , τ, I) locally belongs to I, if A ∩ D τ,I (A) = ∅, i.e., for any x ∈ A there is G ∈ τ containing x such that A ∩ U ∈ I, see Kaniewski et al. (1998). So, X locally belongs to I if and only if for any x ∈ X there is an open set G containing x such that G ∈ I. That means (E, U ,τ ) is soft locally countable (strongly soft locally countable), see Al Ghour and Hamed Worood (2020) if and only if for any F u e ∈ S P(E, U ) there is a set G ∈τ containing F u e such that G ∈Î 0 (Î s ). Since F u e ∈Î 0 (F u e ∈Î s ) for any (e, u) ∈ E × U , then Theorem 10, 29, Corollary 5, 14 of Al Ghour and Hamed Worood (2020) follow from Lemma 4 (2a). Recall that many results of Al Ghour and Hamed Worood (2020) hold for arbitrary soft ideal. In addition toÎ 0 andÎ s , we can consider a soft idealÎ 0 where I 0 = {B ⊂ E ×U : B u is countable for any u ∈ U } and B u = {e ∈ E : (e, u) ∈ B}. The next assertions from Al Ghour and Hamed Worood (2020) follow directly from the above-obtained results. Namely Theorem 2, 3, see Remark 1 (1), Proposition 9, see Remark 1 (1), Theorem 4, 21, see Remark 1 (3), Theorem 5, 22, see Remark 1 (2), Theorem 7, see Theorem 1 (2), Theorem 18, see Remark 1 (1), (5)

Conclusion
This paper contributes to the expanding literature on soft topology. We prove that soft topologies can be characterized by crisp topologies. This is based on bilateral transition that produces soft topologies from crisp topologies and vice versa. Both constructions are explicit and amenable to mathematical manipulations. Various consequences demonstrate that this transition has far reaching implications for the development of soft topology and its extensions.
We have clearly documented the advantage of this bilateral transition in which all notions and results of soft ideal topological spaces have crisp counterparts in ideal topological spaces. This means that the concepts and results that relate to soft ideal topological spaces are fully covered and derivable from standard methods of general topology. From this point of view, we can also evaluate the results from Al Ghour and Hamed Worood (2020) as a copy of known results. Therefore, in further research of soft topological spaces, we propose avoiding the methods and results that are counterparts (consequences) of topological concepts and rather to focus on applications of soft topological spaces.