1. Introduction
The notion of soft set was proposed by Molodtsov, in 1999 (see [1]). After that, the same researcher has applied the concept theory on many different fields and various directions with many other researchers (see [2] [3] [4] [5]). In 2011 and 2012 the notion of soft topological spaces over an initial universe, was studied by many researches (see [6] - [12]). It is known that soft topology is a modern and crucial disciplinary has the ability to the improvement of creative and innovative approaches and new mathematical models that participate in finding solutions of complicated issues in many important sciences. Throughout this study, (X, τ, E) always denotes soft topological space. The concept and the notion of i-open sets in the topological spaces ordinary type were presented by Mohammed, A.A. and Askandar, S.W. in 2012 [13]. These researchers managed to entire these sets together with various generalized open sets concepts. After that, in 2016, the aforementioned results were generalized through the utilization of i-open sets’ notion to present separation axioms i-separation axioms like
,
and
. Also, the researchers managed to find the connection between soft i-separation axioms and different notions like semi-separation axioms, and α-separation axioms. Furthermore, the sets of i-star generalized w-closed and i-open in bitopological spaces
were defined through the utilization of i-open sets the definition which exists in topological space. Also various fundamental relations and properties among these types of groups were introduced. Moreover, the closed sets notion was generalized, by the researchers, to i-generalized closed sets aided by i-open sets and gained topological properties and presented two types of maps known as gi-homeomorphisms and ig-homeomorphisms by examining their properties. However, in this paper, the researches utilize the definitions of i-open sets, α-open sets [14] and semi-open sets [15]. Within the part two 2, the researchers present famous basic concepts and findings of soft topological space and the soft sets’ theory. In Section 3 they will introduce and study soft inter-open, soft i-open and soft ii-open sets’ definitions; then the researchers find its place among some other types of soft open sets as soft semi-open (see [16]) and sets of soft α-open (see [17]) in the soft topological spaces (see Theorems 3.2, 3.5, 3.7, 3.11, 3.12, 3.15, Corollaries 3.4, 3.9, 3.17, 3.20). In Section 4, they initiate and analyze some applications of soft ii-closure, soft ii-open sets as soft ii-interior, soft ii-border, soft ii-frontier and soft ii-exterior of soft sets within the spaces of soft topologies.
2. Preliminaries
Definition 2.1: Consider (X, τ) as the topological space. A subset
will be:
1) [14]. α-open whether
.
2) [15]. Semi-open whether:
a)
.
b) If
such that
.
3) [18]. Inter-open or in short (int-open set) if
such that
.
4) [13]. i-open if
such that
.
5) [18]. ii-open if
and satisfies the both conditions: a)
. b)
. In other words, a subset
is an ii-open if it is inter-open and i-open set together.
Definition 2.2: [1]. Consider X to be a universe’ initial, then P(X) denotes the X set of power; E is a parameters set and (
). A couple (K, A) or KA to be named as a soft set over X, wherein K considered as mapping
. Over X soft set is parameterized collection of subdivisions of X. in Specific
, K(e) considered the group of e-is to be soft set (K, A) group of e-approximate factors, whether “
, so
i.e.
”. SS(XA) denotes all these soft sets over X family.
Definition 2.3: [5]. Consider
. Then KA is a soft subset of LB, denoted by
, whether 1)
. 2)
,
.
Definition 2.4: [5].
are said to be soft equal if
and
.
Definition 2.5: [4]. The complement of (K, A) soft set is designated by (K, A)c, is determined by
,
is a mapping introduced by
, The (K) soft complement function is
and Kc. Apparently,
.
Definition 2.6: [12]. The difference of (K, E) and (L, E) as soft sets in X, indicated by
is the soft set (M, E) wherein,
,
.
Definition 2.7: [12]. Consider (K, E) is a soft set over
and X. It can be said that
whether
,
.
Definition 2.8: [12]. Consider
, as a soft set (x, E) over X, wherein
,
is denoted by xE and was addressed as the single consider on soft point.
Definition 2.9: [5]. (K, A), as a soft set, over X is:
1) A null soft set indicated by
or
whether
,
.
2) An absolute soft set indicated by
or XA if
,
. Clearly
and
.
Definition 2.10: [12]. A soft set
named as a soft point in XE is indicated by
, and eK if
and
,
. The soft point eK belongs to the soft set (G, E),
, whether regarding the factor
,
. The group of X whole soft points is indicated by SP(X).
Definition 2.11: [5]. The soft set (M, C) is the result of the union of both soft sets (L, B) and (K, A) over X wherein
and
Definition 2.12: [5]. The soft set (M, C) is result of the intersection of both soft sets (L, B) and (K, A) on X is wherein
,
,
.
Definition 2.13: [19]. Consider I is an random indexed group and
is subdivision family of SS(XE).
1) (M, E) considered as the soft set of L union wherein
,
.
2) (N, E) considered as the soft set of L crossroad wherein
,
.
Definition 2.14: [20]. Consider the set of attributes be E and the universe is X. Therefore, the soft class which is indicated as (X, E), is the compilation of whole soft sets over X accompanied by attributes from E.
Proposition 2.15: [21]. Consider (K, A),
: therefore,
1)
. 2)
whether and whether
and
. 3)
if and only if
.
Proposition 2.16: [11]. Consider
. Then
1)
. 2)
.
Definition 2.17: [11]. The group of soft sets over X is "τ, the soft topology on X is τ whether:
1)
, XE comes from to τ. 2) τ Owes all the union of number soft sets of τ 3) Also, any 2 soft sets in τ intersection belong to τ. The soft topological space known as the triple consider (X, τ, E). Soft open sets are known as the representative of τ. If the complement (K, E)c of (K, E) in (X, τ, E) belongs to τ, then the soft set called a soft closed set. τc denotes the set of the whole soft closed sets over X.
Definition 2.18: Consider (K, E) be a soft set in (X, τ, E). Therefore,
1) [11]. The intersection of the whole soft closed sets including (K, E) is the soft closure of (K, E) which is indicated by Cl(K, E). Containing (K, E), the lowest soft closed set is Cl(K, E).
2) [8]. The union of the whole soft open sets which are contained in (K, E) is the (K, E) soft interior which is indicated by Int(K, E). Therefore, the largest soft open set contained in (K, E) is Int(K, E).
Theorem 2.19: [8]. Consider (K, E) be a soft set in (X, τ, E). Then
1)
. 2)
. 3)
.
3. Soft ii-Open Sets in the Soft Topological Spaces
Definition 3.1: Consider (F, E) as a soft set in (X, τ, E), therefore, (F, E) is said to be,
1) If there is a soft open set
where
, then (F, E) is soft Inter-open (in short soft Int-open).
2) Soft i-open whether there is a soft open set
where
.
3) Soft ii-open whether there is a soft open set
where:
a)
.
b)
.
4) [16]. Soft semi-open if:
a)
.
b) Whether a soft open set exists
where
.
5) [17]. Soft α-open if
.
The complement of soft ii-open (resp., soft α-open, soft i-open, soft int-open, and soft semi-open) sets known as soft ii-closed (resp., soft int-closed, soft i-closed, soft semi-closed and soft α-closed) sets. The intersection of all soft ii-closed (resp., soft int-closed, soft i-closed, soft semi-closed and soft α-closed) sets over X containing (F, E) is called the soft ii-closure (resp., soft int-closure, soft i-closure, soft semi-closure and soft α-closure) of (F, E) and denoted by ii-Cl(F, E) (resp., int-Cl(F, E), i-Cl(F, E), s-Cl(F, E) and α-Cl(F, E)).
The union of whole soft ii-open (resp., int-open, i-open, semi-open and α-open) sets over X contained in (F, E) known as a soft ii-interior (resp., int-interior, i-interior, semi-interior and α-interior) of a soft set (F, E) and indicated by ii-Int(F, E) (resp., int-Int(F, E), i-Int(F, E), s-Int(F, E) and α-Int(F, E)). The group of whole soft open (resp., ii-open, int-open, i-open, semi-open and α-open sets),(soft closed, ii-closed, int-closed, i-closed, semi-closed and α-closed) sets in (X, τ, E) are indicated by OS(XE) (resp., IIOS(XE), INTOS(XE), IOS(XE), SOS(XE), αOS(XE), CS(XE), IICS(XE), INTCS(XE), ICS(XE), SCS(XE) and αCS(XE)).
Theorem 3.2: Each soft open set is a soft i-open.
Proof: Consider (G, E) as soft open set in (X, τ, E). It is clear that
, then,
. Hence (G, E) is a soft i-open.?
Example 3.3: Consider
,
,
.where
,
.
Consider, “
”.
Apparently,
is a soft i-open set due to the existence a soft open set
where
, yet (F, E) is not soft open.
Corollary 3.4: Every soft closed set is a soft i-closed.
In example (3.3), it is obvious that
is a soft i-closed set but is not soft closed.
Theorem 3.5: Each soft semi-open set is a soft i-open.
Proof: Consider (F, E) as a soft semi-open set in (X, τ, E). by Definition (3.1) (4) (b) there exist a soft open set (G, E) where
(1)
Since
, then
(2)
By (1) and (2) there is,
. Therefore, (F, E) considered as a soft i-open set.?
Example 3.6: Consider
,
,
.
Such that, “
”.
Consider
.
,
.
Apparently (F, E) has been considered as a soft i-open set because of the existence of a soft open set
wherein
, but (F, E) has not been considered soft semi-open.
Theorem 3.7: Each soft α-open set is considered as a soft semi-open.
Proof: Consider (F, E) as soft α-open set in (X, τ, E). In the definitions (3.1) (5) it can be found;
“
,
”, where
is a soft open set for all i. Then
. Thus, (F, E) is a soft semi-open set.?
Example 3.8: Consider “
,
,
”.
Such that
,
,
.
Consider
.
,
“
”
It is apparently that (F, E) considered as a soft semi-open set, yet (F, E) has not been considered a soft α-open because
.
Corollary 3.9: Each soft α-open set is a soft i-open.
Proof: Consider (F, E) as a soft α-open set in (X, τ, E). Based on the theory (3.7) it can be said; (F, E) is a soft semi-open. Based on the theory (3.5) we can find (F, E) is a soft i-open set.?
In Example (3.6), we see that
is a soft i-open set, but is not soft α-open because
.
Corollary 3.10: By Theorems (3.2), (3.5) and (3.7), Corollary (3.9) we have the following diagram as shown in Figure 1.
Theorem 3.11: Each soft semi-open set is a soft ii-open.
Proof: Consider (F, E) as a soft semi-open set in (X, τ, E). Based in the theory (3.5), it can be found that (F, E) is a soft i-open. Such that there is a soft open set (G, E) where
. Now, we shall prove that
. Note that if
, for each soft open set (G,E). Then, “
”. From above inclusions, it concludes that “
”. This means that
. This is a confliction. Thus,
. Therefore, (F, E) has been considered as a soft ii-open set.?
Figure 1. The relation among soft i-open, soft semi-open, soft open, and soft α-open sets.
Theorem 3.12: Each soft open set is a soft ii-open.
Proof: Consider (G, E) as a soft open set in (X, τ, E). Since “
”, it follows that (G, E) is a soft i-open set. Also
, we have (G, E) is a soft Inter-open. Hence (G, E) is a soft ii-open set.?
Example 3.13: Consider
,
,
.
Where
,
.
Consider
.
,
Obviously (F, E) considered as soft ii-open set, yet (F, E) could not be a soft open.
Theorem 3.14: Whether (X, τ, E) considered as a soft topological space, therefore (X, IIO(XE), E) is also soft topological space.
Proof: Consider (F, E) as a soft ii-open set, also consider
. We will prove the existence of a soft open set, say (G, E) wherein,
. As (F, E) considered as soft ii-open set, it pursues that it exists a soft open set
where
and
. Therefore,
implies that
and
. If (G, E) considered as soft closed set, therefore,
and (F, E) can be a soft open set, if (G, E) is not a soft closed. Since
implies that
indicates that
. This creates an opposition. Thus, (F, E) is soft open set. Hence, IIO(XE) satisfies the conditions of soft topology in the Definition (2.17). That means the triple (X, IIO(XE), E) can also be a soft topological space.?
In Example (3.13),
. It is clear that (X, τ, E) and (X, IIO(XE), E) can be a soft topological space.
Theorem 3.15: Each soft α-open set is a soft ii-open.
Proof: Consider (F, E) as a soft α-open set in (X, τ, E). Since
. Thus, (F, E) can be soft semi-open. By Theorem (3.11) it can be found that (F, E) can be taken as a soft ii-open set.?
Example 3.16: Consider
,
,
. Such that
,
,
. Consider
.
,
. Obviously, (F, E) can be classified as a soft ii-open set not as a soft α-open as
.
Corollary 3.17: Each soft α-open set is a soft inter-open.
Proof: Suppose that (F, E) is a soft α-open set. Since each soft α-open set is a soft ii-open (Theorem 3.15) we have (F, E) is a soft ii-open set, and since each soft ii-open set is a soft inter-open (Definition 3.1 (1 and 3)), we have, (F, E) is a soft inter-open set.?
Example 3.18: Consider
,
,
. Such that “
,
”. Consider “
”.
, “
.” Obviously (F, E) can’t be soft α-open but it is a soft inter-open, due to the fact that
.
Remark 3.19: Each soft ii-open set is a soft i-open, soft inter-open, but the contrary is not true, as viewed in example (3.1.18), we see that
is not soft ii-open but is a soft i-open. Also
is not soft ii-open but a soft inter-open.
Corollary 3.20: By Corollaries (3.9) and (3.17), Theorems (3.11), (3.12), (3.15) we have the following diagram as shown in Figure 2.
Figure 2. The relationship among soft ii-open sets and some other concepts of soft open sets.
4. Some Applications of a Soft ii-Open Sets in the Soft Topological Spaces
Definition 4.1: [3]. Consider (K, E) as a soft set over X, and A is a nonempty subdivision set of X. So, the sub-soft set of (K, E) over A indicated by (AK, E), has been known as
,
. Specifically,
.
Definition 4.2: [3] [5]. Consider (F, E) as soft set over X and
, consider A be a nonempty subset of X. We define:
1)
as follows:
.
2)
as follows:
.
3)
if and only if
,
.
4)
as follows:
.
5)
as follows:
.
6) Soft set (F, E) containing a point
, if
.
Definition 4.3: [22]. Consider (G, E) as soft open set in (X, τ, E),
and
. It can be said that (G, E) is e-soft open neighbor of x whether
"Further, a soft open set (G, E) can be considered as a soft open neighbor of a point x whether
,
.
Definition 4.4: Consider (F, E) be a soft set in (X, τ, E). A point
is said to be a limit point of (F, E), if for each soft open neighborhood (G, E) of x we have
. In other words A point
is said to be a limit point of (F, E), if for each soft open set (G, E) containing x we have
. The set of all limit points of (F, E) is called a derived set of (F, E) and denoted by D(F, E). A soft set (F, E) is a soft closed if
such that
,
.
Definition 4.5: A soft ii-open set (G, E) in (X, τ, E) considered as soft ii-open neighbor of
if
.
Definition 4.6: Consider (F, E) as a soft set in (X, τ, E). A point
is an ii-limit point of (F, E) whether for every soft ii-open neighborhood (G, E) of x there is
. In other words, a point
is an ii-limit point of (F, E) whether every soft ii-open set (G, E) which contain x then there is
. The set of whole ii-limit points of (F, E) known as ii-derived set of (F, E) indicated by ii-D(F, E). Note that a point
is not an ii-limit point of (F, E) whether exist a soft ii-open set (G, E) including x where
. A soft set (F, E) is soft ii-closed if
where
,
.
Theorem 4.7: Consider (F, E), (B, E) as 2 soft sets in (X, τ, E), therefore, the next statements hold:
1)
.
2) if
, then
.
3) a)
.
b)
.
4)
.
Proof:
1) As each soft ii-open is a soft open set, it pursue that
.
2) Consider
. Then there exist a soft ii-open set (G, E) containing x where
(1). As
we get
. It implies
, from (1) we have
. Therefore,
. Hence
.
3) a) Since
and
, from (2) we have
and
. This implies to
. b) Since
and
, from (2) we get
and
. Hence
.
4) Consider
, so regarding a soft ii-open set (G, E) containing x, we have
. Thus
or
. It follows that
. Hence
and in any case,
.?
Theorem 4.8: For any soft set (F, E) in (X, τ, E),
.
Proof: Since
and by Theorem (4.7) (2) we have
. Since
considered as soft ii-closed set we have
, therefore,
. Hence
. On the other hand, consider
. If
, this complete the evidence. If
, for each soft ii-open set (G, E) that includes x crossroads (F, E) at a point distinct from x, so
. Thus,
.?
Theorem 4.9: Consider (F, E), (B, E) be two soft sets in (X, τ, E), so the next phrases hold:
1) ii-Int(F, E) is the union of all soft ii-open subsets of (F, E).
2) (F, E) is soft ii-open if and only if
.
3)
.
4)
.
5)
.
6)
.
7) If
, then
.
8)
.
9)
.
Proof:
1) Consider
as a group of whole soft ii-open subdivision sets of (F, E). If
, then there exist
where
. Hence
and so
. Contradict, if
, then
for some
. Thus
and
.
2) Straight forward.
3) It pursues from (1) and (2).
4) If
, then
, so there exist a soft ii-open set (G, E) containing x where
. Thus,
and hence
. Thus shows
. Now consider
. since
belongs to the group of whole soft open sets in X and
, there is
. Therefore,
.
5) Using (4) and Theorem (4.8), we have,
.
6) Using (4) and Theorem (4.8) we get:
.
7) Since
and
,
, we get
.
8) Since
and
, from (7) we get
,
. Therefore,
.
9) Since
and
, from (7) we get
and
. Consequently,
.?
Definition 4.10: Consider (F, E) as soft set within (X, τ, E). We define the soft ii-border of (F, E) indicated by
as showed next:
.
Theorem 4.11: Consider (F, E) as a soft set in (X, τ, E), therefore the next phrases holds:
1)
where b(F, E) indicates the soft border of (F, E) and defined as
.
2)
.
3)
.
4)
whether and only whether (F, E) considered as soft ii-open set.
5)
.
6)
.
7)
.
8)
.
9)
.
Proof:
1) As
, there is,
.
2) Straightforward.
3) Straightforward.
4) Since
, it follows based on the Theorem (4.9) (2), that (F, E) can be a soft ii-open set whether and only whether
whether and only whether 1)
.
5) Since
is a soft ii-open set following from (4) that
.
6) If
, then
. On the other hand, since
,
. Hence,
. Which contradict with (3). Thus
.
7) Using (6) we get
.
8) Using Theorem (4.9) (6), we have
.
9) Applying (8) and Theorem (4.8), we have
.
Definition 4.12: Consider (F, E) as a soft open in (X, τ, E). We define the soft ii-Frontier of (F, E) indicated thru
as pursues:
.
Theorem 4.13: Consider (F, E) as a soft set in (X, τ, E), so the next statements hold:
1)
where Fr(F, E) denotes the soft frontier of (F, E) and known as
.
2)
.
3)
.
4)
.
5)
.
6)
whether and only whether (F, E) considered as a soft ii-open.
7)
.
8)
.
9)
is soft ii-closed set.
10)
.
11)
.
12)
.
13)
.
Proof:
1) Since
and
, it follows that:
.
2)
.
3)
.
4) Since
, we have
.
5) Since
, we have
.
6) Assume (F, E) considered as soft ii-open. Then utilizing Theorem (4.9) (2) and Theorem (4.11) (4) (9), we have,
.
Conversely, suppose that
. Then
. By using (4) and (5) of Theorem (4.9) and so
. Since
in general, they pursue that
. So from Theorem (4.9) (2) (F, E) is considered to be soft ii-open set.
7)
.
8) It follows from (7).
9)
. Hence ii-Fr(F, E) is soft ii-closed.
10)
.
11) Using Theorem (3.2.9) (3), we get
.
12) Since
, we have
.
13)
.?
Definition 4.14: Consider (F, E) as soft set within (X, τ, E). We define the soft ii-Exterior of (F, E) indicated by
as:
.
Theorem 4.15: Consider (F, E) as a soft set within (X, τ, E), so next phrases means:
1)
where Ext(F, E) denotes the soft exterior of (F, E) and known as
.
2)
is soft ii-open set.
3)
.
4)
.
5) If
, then
.
6)
.
7)
.
8)
.
9)
.
10)
.
11)
.
12)
.
Proof:
1) As
, we have,
.
2) It follows from Theorem (4.9) (1).
3) It is straightforward by Theorem (4.9) (6).
4)
.
5) Assume that
, then utilizing Theorem (4.9) (7), we have
.
6) Applying Theorem (4.9) (8), we get
.
7) Applying Theorem (4.9) (9), we obtain
.
8) Straightforward.
9) Straightforward.
10)
.
11)
.
12)
.
Example 4.16: Consider
,
,
. Where
,
,
.
,
.
Soft i-open sets are:
.
Soft ii-open sets are:
.
Soft ii-closed sets are:
.
Consider
,
1)
. Obviously
is considered as soft ii-closed set. Consider
.
2)
. Clearly
is not soft ii-closed set.
3)
.
4)
.
5)
. Clearly
is soft ii-open set.
6)
.
7)
,
.
8)
.
9)
.
10)
,
.
11)
.
12)
.
13)
,
. Hence
.
14)
.
15)
.
16)
, hence (FK, E) is soft ii-open set,
, hence (FL, E) is soft ii-open set,
, hence (F6, E) is not soft ii-open set.
17)
,
.
.
18)
,
.
19)
, hence
.
20)
,
. Hence
.
21)
,
,
, (F6, E) is not soft ii-open set,
,
,
,
, (F6, E) is soft ii-closed set,
,
,
.
22)
,
,
is soft ii-open set,
,
,
,
,
,
, we have
,
,
,
,
,
,
,
, where
.
5. Conclusions
1) Each soft open set is a soft i-open but the converse is not true.
2) Each soft semi-open set is a soft i-open but the converse is not true.
3) Each soft ii-open set is a soft i-open and a soft inter-open set but the converse is not true.
4) Each soft α-open set is a soft ii-open set which implies to: each soft α-open set is a soft i-open and a soft inter-open set but the converses are not true.
Acknowledgements
The author is grateful to Prof. Amir A. Mohammed for his valuable remarks.