Some Local Properties of Soft Semi-Open Sets

Abstract

We introduce some local properties by soft semi-open sets. For example, soft semi-neighborhoods of the soft point, soft semi-first-countable spaces and soft semi-pu-continuous at the soft point are given. Furthermore, we define soft semi-connectedness and prove that a soft topological space is soft semiconnected if and only if both soft semi-open and soft semi-closed sets are only ∅ and 𝑋.

1. Introduction

Soft set theory [1] was firstly introduced by Molodtsov in 1999 as a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, and so forth, in [1].

In the recent years, papers about soft sets theory and their applications in various fields have been writing increasingly [2–6]. Shabir and Naz [7] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. The authors introduced the definitions of soft open sets, soft closed sets, soft interior, soft closure, and soft separation axioms. And the authors got some important results for soft separation axioms. The results are valuable for research in this field. Tanay and Kandemir [8] introduced the definition of fuzzy soft topology over a subset of the initial universe set while Jun et al. [9] studied the ideal theory in BCK/BCI-Algebras based on soft sets. Along the line of Shabir and Naz, Chen introduced the notations of soft semi-open sets in soft topological spaces in [10].

In the present study, we introduce some local properties by soft semi-open sets. For example, soft semi-neighborhoods of the soft point, soft semi-first-countable spaces, and soft semi-𝑝𝑢-continuous at the soft point are given. And some of their properties are studied.

2. Preliminaries

Let 𝑈 be an initial universe set and 𝐸_𝑈 be a collection of all possible parameters with respect to 𝑈, where parameters are the characteristics or properties of objects in 𝑈. We will call 𝐸_𝑈 the universe set of parameters with respect to 𝑈.

Definition 1 (see [1]). A pair (𝐹,𝐴) is called a soft set over 𝑈 if 𝐴⊂𝐸_𝑈 and 𝐹∶𝐴→𝑃(𝑈), where 𝑃(𝑈) is the set of all subsets of 𝑈.

Definition 2 (see [1]). Let 𝑈 be an initial universe set and 𝐸_𝑈 be a universe set of parameters. Let (𝐹,𝐴) and (𝐺,𝐵) be soft sets over a common universe set 𝑈 and 𝐴,𝐵⊂𝐸. Then (𝐹,𝐴) is a subset of (𝐺,𝐵), denoted by (𝐹,𝐴)⊂(𝐺,𝐵), if: (i) 𝐴⊂𝐵; (ii) for all 𝑒∈𝐴, 𝐹(𝑒)⊂𝐺(𝑒).
(𝐹,𝐴) equals (𝐺,𝐵), denoted by (𝐹,𝐴)=(𝐺,𝐵), if (𝐹,𝐴)⊂(𝐺,𝐵) and (𝐺,𝐵)⊂(𝐹,𝐴).

Definition 3 (see [1]). A soft set (𝐹,𝐴) over 𝑈 is called a null soft set, denoted by ∅, if 𝑒∈𝐴, 𝐹(𝑒)=∅.

Definition 4 (see [1]). A soft set (𝐹,𝐴) over 𝑈 is called an absolute soft set, denoted by 𝐴, if 𝑒∈𝐴, 𝐹(𝑒)=𝑈.

Definition 5 (see [1]). The union of two soft sets (𝐹,𝐴) and (𝐺,𝐵) over a common universe 𝑈 is the soft set (𝐻,𝐶), where 𝐶=𝐴∪𝐵, and for all 𝑒∈𝐶, 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒)𝐹(𝑒)(𝑒),if𝑒∈𝐴−𝐵,,𝑒∈𝐵−𝐴,∪𝐺,𝑒∈𝐵∩𝐴. (1)
We write (𝐹,𝐴)∪(𝐺,𝐵)=(𝐻,𝐶).

Definition 6 (see [1]). The intersection of two soft sets of (𝐹,𝐴) and (𝐺,𝐵) over a common universe 𝑈 is the soft set (𝐻,𝐶), where 𝐶=𝐴∩𝐵, and for all 𝑒∈𝐶, 𝐻(𝑒)=𝐹(𝑒)∩𝐺(𝑒). We write (𝐹,𝐴)∩(𝐺,𝐵)=(𝐻,𝐶).
Now we recall some definitions and results defined and discussed in [6, 7, 10]. Henceforth, let 𝑋 be an initial universe set and 𝐸 be the fixed nonempty set of parameter with respect to 𝑋 unless otherwise specified.

Definition 7. For a soft set (𝐹,𝐴) over 𝑈, the relative complement of (𝐹,𝐴) is denoted by (𝐹,𝐴)^′ and is defined by (𝐹,𝐴)^′=(𝐹^′,𝐴), where 𝐹^′∶𝐴→𝑃(𝑈) is a mapping given by 𝐹^′(𝑒)=𝑈−𝐹(𝑒) for all 𝑒∈𝐴.

Definition 8. Let 𝜏 be the collection of soft sets over 𝑋, then 𝜏 is called a soft topology on 𝑋 if 𝜏 satisfies the following axioms.(1)∅,𝑋 belong to 𝜏.(2)The union of any number of soft sets in 𝜏 belongs to 𝜏.(3)The intersection of any two soft sets in 𝜏 belongs to 𝜏.
The triplet (𝑋,𝜏,𝐸) is called a soft topological space over 𝑋.

Definition 9. Let (𝑋,𝜏,𝐸) be a soft space over 𝑋, then the members of 𝜏 are said to be soft open sets in 𝑋.

Definition 10. Let (𝑋,𝜏,𝐸) be a soft space over 𝑋. A soft set (𝐹,𝐸) over 𝑋 is said to be a soft closed set in 𝑋, if its relative complement (𝐹,𝐸)^′ belongs to 𝜏.

Proposition 11. Let (𝑋,𝜏,𝐸) be a soft space over 𝑋. Then one has the following.(1)∅,𝑋 are soft closed sets over 𝑋.(2)The intersection of any number of soft closed sets is a soft closed set over 𝑋.(3)The union of any two soft closed sets is a soft closed set over 𝑋.

Definition 12. Let (𝑋,𝜏,𝐸) be a soft topological space and (𝐴,𝐸) be a soft set over 𝑋.(1)The soft interior of (𝐴,𝐸) is the soft set Int(𝐴,𝐸)=⋃{(𝑂,𝐸)∶(𝑂,𝐸) is soft open and (𝑂,𝐸)⊂(𝐴,𝐸)}.(2)The soft closure of (𝐴,𝐸) is the soft set Cl(𝐴,𝐸)=⋂{(𝐹,𝐸)∶(𝐹,𝐸) is soft closed and (𝐴,𝐸)⊂(𝐹,𝐸)}.
By property 2 for soft open sets, Int(𝐴,𝐸) is soft open. It is the largest soft open set contained in (𝐴,𝐸).
By property 2 for soft closed sets, Cl(𝐴,𝐸) is soft closed. It is the smallest soft closed set containing (𝐴,𝐸).

Proposition 13. Let (𝑋,𝜏,𝐸) be a soft topological space and let (𝐹,𝐸) and (𝐺,𝐸) be a soft set over 𝑋. Then(1)Int(Int(𝐹,𝐸))=Int(𝐹,𝐸),(2)(𝐹,𝐸)⊂(𝐺,𝐸) implies Int(𝐹,𝐸)⊂Int(𝐺,𝐸),(3)Cl(Cl(𝐹,𝐸))=Cl(𝐹,𝐸),(4)(𝐹,𝐸)⊂(𝐺,𝐸) implies Cl(𝐹,𝐸)⊂𝑥Cl(𝐺,𝐸).

Definition 14. A soft set (𝐴,𝐸) in a soft topological space (𝑋,𝜏,𝐸) will be termed soft semi-open (written S.S.O) if and only if there exists a soft open set (𝑂,𝐸) such that (𝑂,𝐸)⊂(𝐴,𝐸)⊂Cl(𝑂,𝐸).

Definition 15. A soft set (𝐵,𝐸) in a soft topological space (𝑋,𝜏,𝐸) will be termed soft semi-closed (written S.S.C) if its relative complement is soft semi-open, for example, there exists a soft closed set (𝐹,𝐸) such that Int(𝐹,𝐸)⊆(𝐵,𝐸)⊂(𝐹,𝐸).

Definition 16. Let (𝑋,𝜏,𝐸) be a soft topological space and (𝐴,𝐸) be a soft set over 𝑋.(1)The soft semi-interior of (𝐴,𝐸) is the soft set Int_𝑆(𝐴,𝐸)=⋃{(𝑂,𝐸)∶(𝑂,𝐸) is soft semi-open and (𝑂,𝐸)⊂(𝐴,𝐸)}.(2)The soft semi-closure of (𝐴,𝐸) is the soft set Cl_𝑆(𝐴,𝐸)=⋂{(𝐹,𝐸)∶(𝐹,𝐸) is soft semi-closed and (𝐴,𝐸)⊂(𝐹,𝐸)}.

3. Soft Semi-Neighborhoods

Definition 17. A soft set (𝐹,𝐸) in a soft topological space (𝑋,𝜏,𝐸) is said to be a soft semi-neighborhood of the soft point 𝑒_𝐹 if there is a soft semi-open set (𝐵,𝐸) s.t. 𝑒_𝐹∈(𝐵,𝐸)⊆(𝐹,𝐸).
The semi-neighborhood system of a soft point 𝑒_𝐹 which is denoted by 𝒰_𝑒𝐹 is the set of all its semi-neighborhoods.

Proposition 18. The semi-neighborhood system 𝒰_𝑒𝐹 at a soft point 𝑒_𝐹 in the soft topological space (𝑋,𝜏,𝐸) has the following results.(a)If (𝐹,𝐸)∈𝒰_𝑒𝐹, then one has 𝑒_𝐹∈(𝐹,𝐸).(b)If (𝐹,𝐸)∈𝒰_𝑒𝐹 and (𝐹,𝐸)⊆(𝐵,𝐸), then one has (𝐵,𝐸)∈𝒰_𝑒𝐹.(c)If (𝐹,𝐸)∈𝒰_𝑒𝐹, then there exists a soft set (𝐵,𝐸)∈𝒰_𝑒𝐹 s.t. (𝐹,𝐸)∈𝒰_𝑒′ for each 𝑒^′∈(𝐵,𝐸).

Proof. (a) If (𝐹,𝐸)∈𝒰_𝑒𝐹, then we have a soft semi-open set (𝐵,𝐸) s.t. 𝑒_𝐹∈(𝐵,𝐸)⊆(𝐹,𝐸). So we have 𝑒_𝐹∈(𝐹,𝐸).
(b) If (𝐹,𝐸)∈𝒰_𝑒𝐹 and (𝐹,𝐸)⊆(𝐵,𝐸). Because (𝐹,𝐸)∈𝒰_𝑒𝐹, there exists a soft semi-open set (𝐻,𝐸) s.t. 𝑒_𝐹∈(𝐻,𝐸)⊆(𝐹,𝐸). So we have 𝑒_𝐹∈(𝐻,𝐸)⊆(𝐹,𝐸)⊆(𝐵,𝐸) and we have (𝐵,𝐸)∈𝒰_𝑒𝐹.
(c) If (𝐹,𝐸)∈𝒰_𝑒𝐹, then there is a soft semi-open set (𝐻,𝐸) s.t. 𝑒_𝐹∈(𝐻,𝐸)⊆(𝐹,𝐸). Let (𝐵,𝐸)=(𝐻,𝐸) and for each 𝑒^′∈(𝐵,𝐸),𝑒^′∈(𝐵,𝐸)⊆(𝐻,𝐸)⊆(𝐹,𝐸). This shows (𝐹,𝐸)∈𝒰_𝑒′.

Definition 19. Let (𝑋,𝜏,𝐸) be a soft topological space and 𝒰_𝑒𝐹 be a soft semi-neighborhood of a soft point 𝑒_𝐹. If for every soft semi-neighborhood (𝐹,𝐸) of 𝑒_𝐹, there is a (𝐻,𝐸)∈𝒱_𝑒𝐹⊆𝒰_𝑒𝐹 such that 𝑒_𝐹∈(𝐻,𝐸)⊆(𝐹,𝐸), then 𝒱_𝑒𝐹 is said to be a soft semi-neighborhoods base of 𝒰_𝑒𝐹 at 𝑒_𝐹.

Proposition 20. Let (𝑋,𝜏,𝐸) be a soft topological space and 𝒰_𝑒𝐹 be a soft semi-neighborhood of a soft point 𝑒_𝐹. 𝒱_𝑒𝐹 is the soft semi-neighborhoods base of 𝒰_𝑒𝐹 at 𝑒_𝐹. Then one has the following.(a)If (𝐹,𝐸)∈𝒱_𝑒𝐹, then we have 𝑒_𝐹∈(𝐹,𝐸).(b)If (𝐹,𝐸)∈𝒱_𝑒𝐹 and (𝐹,𝐸)⊆(𝐵,𝐸), then one has (𝐵,𝐸)∈𝒱_𝑒𝐹.(c)If (𝐹,𝐸)∈𝒱_𝑒𝐹, then there exists a soft set (𝐵,𝐸)∈𝒱_𝑒𝐹 s.t. (𝐹,𝐸)∈𝒱_𝑒′ for each 𝑒^′∈(𝐵,𝐸).

Proof. These properties are easily verified by referring to the corresponding properties of soft seminbds in Proposition 18.

Definition 21. Let (𝑋,𝜏,𝐸) be a soft topological space and 𝑒_𝐹 be a soft point in (𝑋,𝜏,𝐸). If 𝑒_𝐹 has a countable soft semi-neighborhoods base, then we say that (𝑋,𝜏,𝐸) is soft semi-first-countable at 𝑒_𝐹. If each soft point in (𝑋,𝜏,𝐸) is soft semi-first-countable, then we say that (𝑋,𝜏,𝐸) is a soft semi-first-countable space.

Proposition 22. Let (𝑋,𝜏,𝐸) be a soft topological space and 𝑒_𝐹 be a soft point in (𝑋,𝜏,𝐸). Then (𝑋,𝜏,𝐸) is soft semi-first-countable at 𝑒_𝐹 if and only if there is a countable soft semi-neighborhoods base {(𝐹_𝑛,𝐸),𝑛∈𝑁} at 𝑒_𝐹 such that (𝐹_𝑛+1,𝐸)⊆(𝐹_𝑛,𝐸) for each 𝑛∈𝑁.

Proof. ⇐ Obvious.
⇒ Let {(𝑈_𝑛,𝐸),𝑛∈𝑁} be a countable soft semi-neighborhoods base at 𝑒_𝐹. For each 𝑛∈𝑁, put (𝐹_𝑛,𝐸)=⋂^𝑛_𝑖=1(𝑈_𝑖,𝐸). Then it is easy to see that {(𝐹_𝑛,𝐸),𝑛∈𝑁} is a soft semi-neighborhoods base at 𝑒_𝐹 and (𝐹_𝑛+1,𝐸)⊆(𝐹_𝑛,𝐸) for each 𝑛∈𝑁.

Definition 23. Let (𝑋,𝜏,𝐸) be a soft topological space, (𝐵,𝐸) be a soft set, and 𝑒_𝐹 be a soft point in (𝑋,𝜏,𝐸). Then 𝑒_𝐹 is said to be a soft semi-interior point of (𝐵,𝐸) if there is a soft semi-open set (𝐻,𝐸) satisfies 𝑒_𝐹∈(𝐻,𝐸)⊆(𝐵,𝐸).

Proposition 24. Let 𝑒_𝐹 be a soft point in (𝑋,𝜏,𝐸) and (𝐵,𝐸) be a soft semi-open set. Then one has the following results.(a)Each soft point 𝑒_𝐹∈(𝐵,𝐸) is a soft semi-interior point.(b)For each 𝑒∈𝐸, one can define (𝐵,{𝑒}) as follows:𝐵𝑒^′=𝐵(𝑒),𝑖𝑓𝑒^′=𝑒,∅,𝑖𝑓𝑒^′≠𝑒. (2)Then (𝐵,{𝑒}) is a soft semi-interior point of (𝐵,𝐸) and (𝐵,{𝑒})=∪𝑒_𝐹 for each soft semi-interior point 𝑒_𝐹 of (𝐵,𝐸).(c)∪_𝑒∈𝐸(𝐵,{𝑒})=(𝐵,𝐸).

Proof. (a), (c) are obvious by definitions.
(b) For (𝐵,𝐸) is a soft semi-open set, the soft semi-interior point (𝐵,{𝑒}) is the largest soft semi-interior point of (𝐵,𝐸) by 𝑒∈𝐸 and we have (𝐵,{𝑒})=∪𝑒_𝐹 for every soft semi-interior point 𝑒_𝐹 of (𝐵,𝐸).

Proposition 25. Let (𝑋,𝜏,𝐸) be a soft topological space, (𝐵,𝐸) be a soft set. Then Int_𝑆(𝐵,𝐸)=⋃_𝑒∈𝐸{𝑒_𝐹∶𝑒_𝐹 is any soft semi-interior point of (𝐵,𝐸)}.

Proof. By the definition of soft semi-interior and Proposition 24.

Definition 26. A soft set (𝐹,𝐸) in a soft topological space (𝑋,𝜏,𝐸) is called a soft semi-neighborhood (briefly: snbd) of the soft set (𝐺,𝐸) if there is a soft semi-open set (𝐵,𝐸) s.t. (𝐺,𝐸)⊆(𝐵,𝐸)⊆(𝐹,𝐸).

Proposition 27. Let (𝑋,𝜏,𝐸) be a soft topological space, (𝐵,𝐸) be a soft set. Then (𝐵,𝐸) is soft semi-open if and only if for each soft set (𝐹,𝐸) contained in (𝐵,𝐸), (𝐵,𝐸) is a soft semi-nbd of (𝐹,𝐸).

Proof. ⇒ obvious
⇐ Because (𝐵,𝐸)⊆(𝐵,𝐸), then we have a soft semi-open set (𝐻,𝐸) s.t. (𝐵,𝐸)⊆(𝐻,𝐸)⊆(𝐵,𝐸). So we have (𝐵,𝐸)=(𝐻,𝐸) and (𝐵,𝐸) is soft semi-open.

Definition 28 (see [6]). Let the sequence {(𝐹_𝑛,𝐸),𝑛∈𝑁} be a soft sequence in a soft topological space (𝑋,𝜏,𝐸). Then {(𝐹_𝑛,𝐸),𝑛∈𝑁} is eventually contained in a soft set (𝐵,𝐸) if and only if there is an integer 𝑚 such that, if 𝑛≥𝑚, then (𝐹_𝑛,𝐸)⊆(𝐵,𝐸). The sequence is frequently contained in (𝐵,𝐸) if and only if for each integer 𝑚, there is an integer 𝑛 such that 𝑛≥𝑚 and (𝐹_𝑛,𝐸)⊆(𝐵,𝐸).

Definition 29. Let the sequence {(𝐹_𝑛,𝐸),𝑛∈𝑁} be a soft sequence in a soft topological space (𝑋,𝜏,𝐸), then one says {(𝐹_𝑛,𝐸),𝑛∈𝑁} semiconverges to a soft point 𝑒_𝐹 if it is eventually contained in each semi-nbd of 𝑒_𝐹. And the soft point 𝑒_𝐹 is said to be a semicluster soft point of {(𝐹_𝑛,𝐸),𝑛∈𝑁} if the sequence is frequently contained in every semi-nbd of 𝑒_𝐹.

Proposition 30. Let (𝑋,𝜏,𝐸) be soft semi-first-countable, then one has the following.(a)(𝐵,𝐸) is soft semi-open ⇔ for every soft sequence {(𝐹_𝑛,𝐸),𝑛∈𝑁} which semiconverges to 𝑒_𝐹 in (𝐵,𝐸) is eventually contained in (𝐵,𝐸).(b)If 𝑒_𝐹 is a semicluster soft point of the soft sequence {(𝐹_𝑛,𝐸),𝑛∈𝑁}, then one has a subsequence of {(𝐹_𝑛,𝐸),𝑛∈𝑁} which semi-converges to 𝑒_𝐹.

Proof. (a) ⇒ Because (𝐵,𝐸) is soft semi-open, (𝐵,𝐸) is a semi-nbd of 𝑒_𝐹, and {(𝐹_𝑛,𝐸),𝑛∈𝑁} semi-converges to 𝑒_𝐹. Then we have {(𝐹_𝑛,𝐸),𝑛∈𝑁} is eventually contained in (𝐵,𝐸).
⇐ For each 𝑒_𝐹 contained in (𝐵,𝐸), let {(𝐹_𝑛,𝐸),𝑛∈𝑁} be the semi-nbd systems such that (𝐹_𝑛+1,𝐸)⊆(𝐹_𝑛,𝐸) for each 𝑛∈𝑁 by Proposition 22. Then {(𝐹_𝑛,𝐸),𝑛∈𝑁} is eventally contained in each semi-nbd of 𝑒_𝐹 that is, {(𝐹_𝑛,𝐸),𝑛∈𝑁} semi-converges to 𝑒_𝐹. So we have an integer 𝑚 such that, if 𝑛≥𝑚, (𝐹_𝑛,𝐸)⊆(𝐵,𝐸). Then (𝐵,𝐸) is a semi-nbd of 𝑒_𝐹, and by Proposition 27, (𝐵,𝐸) is soft semi-open.
(b) Let {(𝐹_𝑛,𝐸)},𝑛∈𝑁 be the seminbds such that (𝐹_𝑛+1,𝐸)⊆(𝐹_𝑛,𝐸) for each 𝑛∈𝑁 by Proposition 22. For every nonnegative integer 𝑖, find 𝑓(𝑖) satisfies 𝑓(𝑖)≥𝑖 and (𝐹_𝑓(𝑖),𝐸)⊆(𝐹_𝑖,𝐸). Then {(𝐹_𝑓(𝑖),𝐸),𝑖∈𝑁} is a subsequence of the sequence {(𝐹_𝑛,𝐸),𝑛∈𝑁}. Obviously this subsequence semiconverges to 𝑒_𝐹.

4. Soft Semi-𝑝𝑢-Continuous Functions

In this part, we define the soft semi-𝑝𝑢-continuous functions induced by two mapping 𝑢∶𝑋→𝑌 and 𝑝∶𝐸₁→𝐸₂ on soft topological spaces (𝑋,𝜏₁,𝐸₁), (𝑌,𝜏₂,𝐸₂) and study some properties on them. We use 𝑆𝑆(𝑋)_𝐸1 and 𝑆𝑆(𝑌)_𝐸2 denote all soft sets on (𝑋,𝐸₁) and (𝑌,𝐸₂). 𝑆𝑆𝑆(𝑋)_𝐸1 and 𝑆𝑆𝑆(𝑌)_𝐸2 denote all semi-open soft sets on soft topological spaces (𝑋,𝜏₁,𝐸₁) and (𝑌,𝜏₂,𝐸₂).

Definition 31. Let (𝑋,𝜏₁,𝐸₁) and (𝑌,𝜏₂,𝐸₂) be two soft topological spaces. Let 𝑢∶𝑋→𝑌 and 𝑝∶𝐸₁→𝐸₂ be mappings. Let 𝑓_𝑝𝑢∶𝑆𝑆(𝑋)_𝐸1→𝑆𝑆(𝑌)_𝐸2 be a function and 𝑒_𝐹 be a soft point in (𝑋,𝜏₁,𝐸₁).(a)𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous at 𝑒_𝐹 if for any (𝐵,𝐸₂)∈𝒰^𝜏2_{𝑓𝑝𝑢(𝑒𝐹)}, there is a (𝐹,𝐸₁)∈𝒰^𝜏1_𝑒𝐹 s.t. 𝑓_𝑝𝑢((𝐹,𝐸₁))⊆(𝐵,𝐸₂).(b)𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous from (𝑋,𝜏₁,𝐸₁) to (𝑌,𝜏₂,𝐸₂) if 𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous for every soft point of (𝑋,𝜏₁,𝐸₁).

Proposition 32. Let (𝑋,𝜏₁,𝐸₁) and (𝑌,𝜏₂,𝐸₂) be two soft topological spaces. Let 𝑢∶𝑋→𝑌 and 𝑝∶𝐸₁→𝐸₂ be mappings. Let 𝑓_𝑝𝑢∶𝑆𝑆(𝑋)_𝐸1→𝑆𝑆(𝑌)_𝐸2 be a function and 𝑒_𝐹 be a soft point in (𝑋,𝜏₁,𝐸₁). Then the following statements are equal.(a)𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous at 𝑒_𝐹.(b)For any (𝐵,𝐸₂)∈𝒰^𝜏2_{𝑓𝑝𝑢(𝑒𝐹)}, there is a (𝐹,𝐸₁)∈𝒰^𝜏1_𝑒𝐹 satisfies (𝐹,𝐸₁)⊆𝑓⁻¹_𝑝𝑢(𝐵,𝐸₂).(c)For any (𝐵,𝐸₂)∈𝒰^𝜏2_{𝑓𝑝𝑢(𝑒𝐹)}, 𝑓⁻¹_𝑝𝑢(𝐵,𝐸₂)∈𝒰^𝜏1_𝑒𝐹.

Proof. From Definition 31, this is obvious.

Proposition 33. Let (𝑋,𝜏₁,𝐸₁) and (𝑌,𝜏₂,𝐸₂) be two soft topological spaces. Let 𝑢∶𝑋→𝑌 and 𝑝∶𝐸₁→𝐸₂ be mappings. 𝑓_𝑝𝑢∶𝑆𝑆(𝑋)_𝐸1→𝑆𝑆(𝑌)_𝐸2 be a function. Then the following statements are equal.(a)𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous.(b)If (𝐵,𝐸₂)∈𝑆𝑆𝑆(𝑌)_𝐸2, one has 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))∈𝑆𝑆𝑆(𝑋)_𝐸1.(c)For each soft semi-closed set (𝐹,𝐸₂)∈(𝑌,𝜏₂,𝐸₂), one has 𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)) is soft semi-closed in (𝑋,𝜏₁,𝐸₁).

Proof. (a) ⇒ (b) Let (𝐵,𝐸₂)∈𝑆𝑆𝑆(𝑌)_𝐸2 and 𝑒_𝐹∈𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)). Next, we will prove that 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))∈𝒰^𝜏1_𝑒𝐹. Because 𝑓_𝑝𝑢(𝑒_𝐹)∈(𝐵,𝐸₂) and (𝐵,𝐸₂)∈𝑆𝑆𝑆(𝑌)_𝐸2, we have (𝐵,𝐸₂)∈𝒰^𝜏2_{𝑓𝑝𝑢(𝑒𝐹)}. Because 𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous at 𝑒_𝐹, there is a (𝐹,𝐸₁)∈𝒰^𝜏1_𝑒𝐹 s.t. 𝑓_𝑝𝑢((𝐹,𝐸₁))⊆(𝐵,𝐸₂). So we have 𝑒_𝐹∈(𝐹,𝐸₁)⊆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)) and 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))∈𝒰^𝜏1_𝑒𝐹. By Proposition 27, 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))∈𝑆𝑆𝑆(𝑋)_𝐸1.
(b) ⇒ (a) Let 𝑒_𝐹 be a soft point, (𝐵,𝐸₂)∈𝒰^𝜏2_{𝑓𝑝𝑢(𝑒𝐹)}. Then we can get a soft semi-open set (𝐹,𝐸₂)∈𝑆𝑆𝑆(𝑌)_𝐸2 s.t. 𝑓_𝑝𝑢(𝑒_𝐹)∈(𝐹,𝐸₂)⊆(𝐵,𝐸₂). By (b) 𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂))∈𝑆𝑆𝑆(𝑋)_𝐸1 and 𝑒_𝐹∈𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂))⊆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)). And we have 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))∈𝒰^𝜏1_𝑒𝐹. So, we prove that 𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous at each soft point.
(b) ⇒ (c) Let (𝐹,𝐸₂) be soft semi-closed in (𝑌,𝜏₂,𝐸₂). Then (𝐹,𝐸₂)^′∈𝑆𝑆𝑆(𝑌)_𝐸2 and by (b), 𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)^′)∈𝑆𝑆𝑆(𝑋)_𝐸1. Because 𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)^′)=(𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)))^′, we prove 𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)) is soft semi-closed in (𝑋,𝜏₁,𝐸₁).
(c) ⇒ (b) Using the same method in (b) ⇒ (c), we can get the result.

Proposition 34. Let (𝑋,𝜏₁,𝐸₁) and (𝑌,𝜏₂,𝐸₂) be two soft topological spaces. Let 𝑢∶𝑋→𝑌 and 𝑝∶𝐸₁→𝐸₂ be mappings. 𝑓_𝑝𝑢∶𝑆𝑆(𝑋)_𝐸1→𝑆𝑆(𝑌)_𝐸2 be a function. If (𝑋,𝜏,𝐸) is soft semi-first-countable, then the following statements are equal.(a)𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous.(b)If (𝐵,𝐸₁)∈(𝑋,𝜏₁,𝐸₁), the inverse image of any semi-nbd of 𝑓_𝑝𝑢((𝐵,𝐸₁)) is a semi-nbd of (𝐵,𝐸₁).(c)If (𝐵,𝐸₁)∈(𝑋,𝜏₁,𝐸₁) and for every semi-nbd (𝐹,𝐸₂) of 𝑓_𝑝𝑢((𝐵,𝐸₁)), there exists a semi-nbd (𝐺,𝐸₁) of (𝐵,𝐸₁) satisfies 𝑓_𝑝𝑢((𝐺,𝐸₁))⊆(𝐹,𝐸₂).(d)Let the soft sequence {(𝐹_𝑛,𝐸₁),𝑛∈𝑁} semi-converges to 𝑒_𝐹, then one has {𝑓_𝑝𝑢(𝐹_𝑛,𝐸₁),𝑛∈𝑁} semi-converges to 𝑓_𝑝𝑢(𝑒_𝐹).

Proof. (a) ⇒ (b) Let 𝑓_𝑝𝑢 be soft semi-𝑝𝑢-continuous. If (𝐹,𝐸₂) is a semi-nbd of 𝑓_𝑝𝑢((𝐵,𝐸₁)), then (𝐹,𝐸₂) contains a soft semi-open nbd (𝐻,𝐸₂) of 𝑓_𝑝𝑢((𝐵,𝐸₁)). Because 𝑓_𝑝𝑢((𝐵,𝐸₁))⊆(𝐻,𝐸₂)⊆(𝐹,𝐸₂), we have 𝑓⁻¹_𝑝𝑢(𝑓_𝑝𝑢((𝐵,𝐸₁)))⊆𝑓⁻¹_𝑝𝑢((𝐻,𝐸₂))⊆𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)). Since (𝐵,𝐸₁)⊆𝑓⁻¹_𝑝𝑢(𝑓_𝑝𝑢((𝐵,𝐸₁))) and 𝑓⁻¹_𝑝𝑢((𝐻,𝐸₂)) is soft semi-open. So, we have 𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)) is a semi-nbd of (𝐵,𝐸₁).
(b) ⇒ (c) Let (𝐵,𝐸₁)∈(𝑋,𝜏₁,𝐸₁) be a soft set and (𝐹,𝐸₂) be any semi-nbd of 𝑓_𝑝𝑢((𝐵,𝐸₁)). Then by (b),𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)) is a semi-nbd of (𝐵,𝐸₁). So there is a soft semi-open set (𝐺,𝐸₁)∈(𝑋,𝜏₁,𝐸₁) s.t. (𝐵,𝐸₁)⊆(𝐺,𝐸₁)⊆𝑓⁻¹_𝑝𝑢((𝐹,𝐸₂)). So we have 𝑓_𝑝𝑢((𝐺,𝐸₁))⊆(𝐹,𝐸₂).
(c) ⇒ (d) Let (𝐵,𝐸₂) be a semi-nbd of {𝑓_𝑝𝑢(𝑒_𝐹)}, then there exists a semi-nbd (𝐻,𝐸₁) of 𝑒_𝐹 s.t. 𝑓_𝑝𝑢((𝐻,𝐸₁))⊆(𝐵,𝐸₂). Since the sequence of soft sets {(𝐹_𝑛,𝐸₁),𝑛∈𝑁} semi-converges to 𝑒_𝐹, there is an 𝑚 such that 𝑛≥𝑚, for the semi-nbd (𝐻,𝐸₁) of 𝑒_𝐹, (𝐹_𝑛,𝐸₁)⊆(𝐻,𝐸₁). So we have 𝑓_𝑝𝑢((𝐹_𝑛,𝐸₁))⊆𝑓_𝑝𝑢((𝐻,𝐸₁))⊆(𝐵,𝐸₂). Thus, {𝑓_𝑝𝑢(𝐹_𝑛,𝐸₁),𝑛∈𝑁} semi-converges to 𝑓_𝑝𝑢(𝑒_𝐹).
(d) ⇒ (a) Let (𝐻,𝐸₂) be any semi-open soft set over (𝑌,𝜏₂,𝐸₂). We show 𝑓⁻¹_𝑝𝑢((𝐻,𝐸₂)) is semi-open soft set over (𝑋,𝜏₁,𝐸₁). Let 𝑒_𝐹 be any soft point of 𝑓⁻¹_𝑝𝑢((𝐻,𝐸₂)). Because (𝑋,𝜏,𝐸) is semi-soft first-countable, then there exists a countable soft semi-neighborhoods base {(𝐹_𝑛,𝐸),𝑛∈𝑁} at 𝑒_𝐹 such that (𝐹_𝑛+1,𝐸)⊆(𝐹_𝑛,𝐸) for each 𝑛∈𝑁. Because {(𝐹_𝑛,𝐸),𝑛∈𝑁} is decreasing, there exists an 𝑚 such that for 𝑛≥𝑚, (𝐹_𝑛,𝐸)⊆𝑓⁻¹_𝑝𝑢((𝐻,𝐸₂)). We know each (𝐹_𝑛,𝐸) is soft semi-neighborhood of 𝑒_𝐹, so 𝑓⁻¹_𝑝𝑢((𝐻,𝐸₂)) is a soft semi-neighborhood of 𝑒_𝐹 by Proposition 18 (b). This complete the proof.

Proposition 35. Let (𝑋,𝜏₁,𝐸₁) and (𝑌,𝜏₂,𝐸₂) be two soft topological spaces. Let 𝑢∶𝑋→𝑌 and 𝑝∶𝐸₁→𝐸₂ be mappings. 𝑓_𝑝𝑢∶𝑆𝑆(𝑋)_𝐸1→𝑆𝑆(𝑌)_𝐸2 be a function. Then the following statements are equal.(a)𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous.(b)If (𝐴,𝐸₁)∈(𝑋,𝜏₁,𝐸₁), one has 𝑓_𝑝𝑢(Cl_𝑆(𝐴,𝐸₁))⊂Cl_𝑆𝑓_𝑝𝑢((𝐴,𝐸₁)).(c)If (𝐵,𝐸₂)∈(𝑌,𝜏₂,𝐸₂), one has Cl_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))⊂𝑓⁻¹_𝑝𝑢(Cl_𝑆(𝐵,𝐸₂)).(d)If (𝐵,𝐸₂)∈(𝑌,𝜏₂,𝐸₂), one has 𝑓⁻¹_𝑝𝑢(Int_𝑆(𝐵,𝐸₂))⊂Int_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)).

Proof. (a) ⇒ (b) For 𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous, then by Proposition 33 (c), 𝑓⁻¹_𝑝𝑢(Cl_𝑆𝑓_𝑝𝑢((𝐴,𝐸₁))) is soft semi-closed containing (𝐴,𝐸₁), and thus Cl_𝑆(𝐴,𝐸₁)⊂𝑓⁻¹_𝑝𝑢(Cl_𝑆𝑓_𝑝𝑢((𝐴,𝐸₁))), which gives 𝑓_𝑝𝑢(Cl_𝑆(𝐴,𝐸₁))⊂𝑓_𝑝𝑢𝑓⁻¹_𝑝𝑢(Cl_𝑆𝑓_𝑝𝑢((𝐴,𝐸₁)))⊂Cl_𝑆𝑓_𝑝𝑢 ((𝐴,𝐸₁)).
To prove that (b) ⇒ (c), we apply (b) to (𝐴,𝐸₁)=𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)) and we obtain the inclusion 𝑓_𝑝𝑢(Cl_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)))⊂Cl_𝑆𝑓_𝑝𝑢(𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)))⊂Cl_𝑆(𝐵,𝐸₂), which gives Cl_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))⊂𝑓⁻¹_𝑝𝑢(Cl_𝑆(𝐵,𝐸₂)).
To prove that (c) ⇒ (d), we apply (c) to (𝐵,𝐸₂)^′ and we obtain the inclusion Cl_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)^′)⊂𝑓⁻¹_𝑝𝑢(Cl_𝑆(𝐵,𝐸₂)^′), which gives 𝑓⁻¹_𝑝𝑢(Int_𝑆(𝐵,𝐸₂))=𝑓⁻¹_𝑝𝑢((Cl_𝑆(𝐵,𝐸₂)^′)^′)=(𝑓⁻¹_𝑝𝑢(Cl_𝑆(𝐵,𝐸₂)^′))^′⊂(Cl_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)^′))^′=(Cl_𝑆((𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)))^′))^′=Int_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)).
To complete (d) ⇒ (a), for every (𝐵,𝐸₂)∈𝑆𝑆𝑆(𝑌)_𝐸2 we have (𝐵,𝐸₂)=Int_𝑆(𝐵,𝐸₂), and it follows from (d) that 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))⊂Int_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)). Thus, we have 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂))=Int_𝑆𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)), that is, 𝑓⁻¹_𝑝𝑢((𝐵,𝐸₂)) is soft semi-open in (𝑋,𝜏₁,𝐸₁). Then 𝑓_𝑝𝑢 is soft semi-𝑝𝑢-continuous.

5. Soft Semi-Connectedness

Definition 36. Let (𝑋,𝜏,𝐸) be a soft topological space. A soft semiseparation on 𝑋 is a pair (𝐴,𝐸) and (𝐵,𝐸) of nonnull soft semi-open sets s.t. 𝑋=(𝐴,𝐸)⋃(𝐵,𝐸), (𝐴,𝐸)⋂(𝐵,𝐸)=∅.

Definition 37. A soft topological space (𝑋,𝜏,𝐸) is called soft semi-connected space if there is no soft semiseparations on 𝑋. And if (𝑋,𝜏,𝐸) has such soft semiseparations, then (𝑋,𝜏,𝐸) is called soft semi-disconnected space.

Theorem 38. Let (𝑋,𝜏,𝐸) be a soft topological space. (𝐴,𝐸) and (𝐵,𝐸) are semiseparations on 𝑋. If (𝐹,𝐸) is a soft semi-connected subspace of (𝑋,𝜏,𝐸), then one has (𝐹,𝐸)⊆(𝐴,𝐸) or (𝐹,𝐸)⊆(𝐵,𝐸).

Proof. Because (𝐴,𝐸) and (𝐵,𝐸) are soft semi-open sets, then we have (𝐹,𝐸)∩(𝐴,𝐸) and (𝐹,𝐸)∩(𝐵,𝐸) are also soft semi-open sets. Hence (𝐹,𝐸)∩(𝐴,𝐸) and (𝐹,𝐸)∩(𝐵,𝐸) are semiseparations of (𝐹,𝐸). And this is a contradiction. So, one of (𝐹,𝐸)∩(𝐴,𝐸) and (𝐹,𝐸)∩(𝐵,𝐸) is ∅ and thus (𝐹,𝐸)⊆(𝐴,𝐸) or (𝐹,𝐸)⊆(𝐵,𝐸).

Theorem 39. Let (𝑋,𝜏,𝐸) be a soft topological space and (𝐹,𝐸) is a soft semi-connected subspace of (𝑋,𝜏,𝐸). If (𝐹,𝐸)⊆(𝐺,𝐸)⊆Cl_𝑆(𝐹,𝐸), then (𝐺,𝐸) is soft semi-connected.

Proof. Suppose (𝐺,𝐸) is not soft semi-connected, then there exist nonnull soft semi-open sets (𝐴,𝐸) and (𝐵,𝐸) which form a soft semiseparation of (𝐺,𝐸). Then by Theorem 38, we have (𝐹,𝐸)⊆(𝐴,𝐸) or (𝐹,𝐸)⊆(𝐵,𝐸). Suppose (𝐹,𝐸)⊆(𝐴,𝐸), then we have (𝐺,𝐸)⊆Cl_𝑆(𝐹,𝐸)⊆Cl_𝑆(𝐴,𝐸). So (𝐺,𝐸)∩(𝐵,𝐸)⊆Cl_𝑆(𝐴,𝐸)∩(𝐵,𝐸)=(𝐴,𝐸)∩(𝐵,𝐸)=∅. Hence (𝐵,𝐸)=(𝐵,𝐸)∩(𝐺,𝐸)=∅, a contradiction. Thus, we have (𝐺,𝐸) is soft semi-connected.

Theorem 40. A soft topological space (𝑋,𝜏,𝐸) is soft semi-connected if and only if the both soft semi-open and soft semi-closed soft sets are only ∅ and 𝑋.

Proof. ⇒ Let the soft topological space (𝑋,𝜏,𝐸) be soft semi-connected. If (𝐴,𝐸) is both soft semi-open and soft semi-closed in (𝑋,𝜏,𝐸) which is different from ∅ and 𝑋. So (𝐴,𝐸)^′ is a soft semi-open set in (𝑋,𝜏,𝐸) which is different from ∅ and 𝑋. Then (𝐴,𝐸) and (𝐴,𝐸)^′ is a soft semiseparation of (𝑋,𝜏,𝐸). This is a contradiction. So the both soft semi-open and soft semi-closed soft sets are only ∅ and 𝑋.
⇐ Let (𝐴,𝐸) and (𝐵,𝐸) be a soft semiseparation of (𝑋,𝜏,𝐸). Let (𝐴,𝐸)≠𝑋 and by Definition 36(𝐴,𝐸)=(𝐵,𝐸)^′. This proves that (𝐴,𝐸) is both soft semi-open and soft semi-closed in (𝑋,𝜏,𝐸) which is different from ∅ and 𝑋. This is a contradiction. So (𝑋,𝜏,𝐸) is soft semi-connected.

Corollary 41. A soft topological space (𝑋,𝜏₁,𝐸₁) is soft semi-disconnected if and only if there exists a nonnull proper soft subset which is both soft semi-open and semi-closed.

Proof. By Theorem 40.

Theorem 42. Let (𝑋,𝜏₁,𝐸₁) and (𝑌,𝜏₂,𝐸₂) be two soft topological spaces. 𝑓_𝑝𝑢∶𝑆𝑆(𝑋)_𝐸1→𝑆𝑆(𝑌)_𝐸2 be a soft semi-𝑝𝑢-continuous function. If (𝑋,𝜏₁,𝐸₁) is soft semi-connected, then (𝑌,𝜏₂,𝐸₂) is soft semi-connected.

Proof. Suppose that (𝑌,𝜏₂,𝐸₂) is not soft semi-connected. Then there is a soft semiseparation (𝐴,𝐸₂) and (𝐵,𝐸₂) of (𝑌,𝜏₂,𝐸₂). So we have 𝑋=𝑓⁻¹_𝑝𝑢((𝐴,𝐸₂)⋃(𝐵,𝐸₂))=𝑓⁻¹_𝑝𝑢(𝐴,𝐸₂)⋃𝑓⁻¹_𝑝𝑢(𝐵,𝐸₂) and 𝑓⁻¹_𝑝𝑢(𝐴,𝐸₂)∩𝑓⁻¹_𝑝𝑢(𝐵,𝐸₂)=𝑓⁻¹_𝑝𝑢(𝑌)=∅_𝑋. Obviously, 𝑓⁻¹_𝑝𝑢(𝐴,𝐸₂) and 𝑓⁻¹_𝑝𝑢(𝐵,𝐸₂) are different from ∅_𝑋. So 𝑓⁻¹_𝑝𝑢(𝐴,𝐸₂) and 𝑓⁻¹_𝑝𝑢(𝐵,𝐸₂) are a soft semiseparation of (𝑋,𝜏₁,𝐸₁). This forms a contradiction. Thus, (𝑌,𝜏₂,𝐸₂) is soft semi-connected.

6. Concluding Remarks

In this paper, we introduce the concept of soft semi-neighborhoods of the soft point, soft semi-𝑝𝑢-continuous at the soft point and soft semi-connectedness. And some of their properties are studied. In the study, soft semi-first-countable space is also given. And in the following papers, soft Frechét space, soft sequential space, soft set-Frechét space and the tightness of a soft point and their connection with soft semi-first-countable space need further study. We hope that the results in this paper will be useful in practical life and nature society.

Acknowledgments

This work described here is supported by the Grants from the National Natural Science Foundation of China (NSFC no. 10971186, 11061004, 71140004, 61070241, 11226265), the Natural Scientific Foundation of Shandong Province (ZR2010AM019, ZR2010FM035, ZR2011AQ015, ZR2010AQ012, and 2012BSB01159). This work is also supported by the international cooperation in training projects for outstanding young teachers of higher institutions of Shandong Province. The author is grateful to the anonymous referees for careful checking of the details and for helpful comments that improved this paper.