Abstract
In this paper, a new class of generalized soft open sets in soft topological spaces, called soft bopen sets, is introduced and studied. Then discussed the relationships among soft \mathit{\alpha}open sets, soft semiopen sets, soft preopen sets and soft \mathit{\beta}open sets. We also investigated the concepts of soft bopen functions and soft bcontinuous functions and discussed their relations with soft continuous and other weaker forms of soft continuous functions.
Introduction and preliminaries
Molodtsov [1] initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty. He successfully applied the soft set theory into several directions such as smoothness of functions, game theory, Riemann Integration, theory of measurement, and so on. Soft set theory and its applications have shown great development in recent years. This is because of the general nature of parametrization expressed by a soft set. Shabir and Naz [2] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. Later, Zorlutuna et al.[3], Aygunoglu and Aygun [4] and Hussain et al are continued to study the properties of soft topological space. They got many important results in soft topological spaces. Weak forms of soft open sets were first studied by Chen [5]. He investigated soft semiopen sets in soft topological spaces and studied some properties of it. Arockiarani and Arokialancy are defined soft \mathit{\beta}open sets and continued to study weak forms of soft open sets in soft topological space. Later, Akdag and Ozkan [6] defined soft \mathit{\alpha}open (soft \mathit{\alpha}closed) sets.
In the present paper, we introduce some new concepts in soft topological spaces such as soft bopen sets, soft bclosed sets, soft binterior, soft bclosure, soft bcontinuous functions, soft bopen functions and soft bclosed functions. We also study the relationships among soft continuity [7], soft \mathit{\alpha}continuity [6], soft semicontinuity [8], soft precontinuity [6], soft \mathit{\beta}continuous [9] and soft bcontinuity of functions defined on soft topological spaces. With the help of counter examples we show the noncoincidence of these various types of mappings.
Throughout the paper, the space X and Y stand for soft topological spaces with ((X,\mathit{\tau},E) and (Y,v,K)) assumed unless otherwise stated. Moreover, throughout this paper, a soft mapping f:X\to Y stands for a mapping, where f:(X,\mathit{\tau},E)\to (Y,\mathit{\upsilon},K), u:X\to Y and p:E\to K are assumed mappings unless otherwise stated.
Definition 1
[1] Let X be an initial universe and E be a set of parameters. Let P(X) denote the power set of X and A be a nonempty subset of E. A pair (F,A) is called a soft set over X, where F is a mapping given by F:A\to P(X). In other words, a soft set over X is a parameterized family of subsets of the universe X. For \mathit{\epsilon}\in A,F(\mathit{\epsilon}) may be considered as the set of \mathit{\epsilon}approximate elements of the soft set (F,A).
Definition 2
[10] A soft set (F,A) over X is called a null soft set, denoted by \mathrm{\Phi}, if e\in A, F(e)=\mathrm{\varnothing}.
Definition 3
[10] A soft set (F,A) over X is called an absolute soft set, denoted by \stackrel{~}{A}, if e\in A, F(e)=X.
If A=E, then the Auniversal soft set is called a universal soft set, denoted by \stackrel{~}{X}.
Definition 4
[2] Let Y be a nonempty subset of X, then \stackrel{~}{Y} denotes the soft set (Y,E) over X for which Y(e)=Y, for all e\in E.
Definition 5
[10] The union of two soft sets of (F,A) and (G,B) over the common universe X is the soft set (H,C), where C=A\stackrel{~}{\cup}B and for all e\in C,
We write (F,A)\stackrel{~}{\cup}(G,B)=(H,C).
Definition 6
[10] The intersection (H,C) of two soft sets (F,A) and (G,B) over a common universe X, denoted (F,A)\stackrel{~}{\cap}(G,B), is defined as C=A\cap B, and H(e)=F(e)\cap G(e) for all e\in C.
Definition 7
[10] Let (F,A) and (G,B) be two soft sets over a common universe X. (F,A)\stackrel{~}{\subset}(G,B), if A\subset B, and H(e)=F(e)\subset G(e) for all e\in A.
Definition 8
[2] Let \mathit{\tau} be the collection of soft sets over X, then \mathit{\tau} is said to be a soft topology on X if satisfies the following axioms.

(1)
\mathrm{\Phi},\stackrel{\sim}{X} belong to \mathit{\tau},

(2)
the union of any number of soft sets in \mathit{\tau} belongs to \mathit{\tau},

(3)
the intersection of any two soft sets in \mathit{\tau} belongs to \mathit{\tau}.
The triplet (X,\mathit{\tau},E) is called a soft topological space over X. Let (X,\mathit{\tau},E) be a soft topological space over X, then the members of \mathit{\tau} are said to be soft open sets in X. A soft set (F,A) over X is said to be a soft closed set in X, if its relative complement {(F,A)}^{c} belongs to \mathit{\tau}.
Definition 9
[11] For a soft set (F,A) over X, the relative complement of (F,A) is denoted by {(F,A)}^{c} and is defined by {(F,A)}^{c}=({F}^{c},A), where {F}^{c}:A\to P(X) is a mapping given by {F}^{c}(\mathit{\alpha})=XF(\mathit{\alpha}) for all \mathit{\alpha}\in A.
Soft bopen sets
In this section we introduce soft bopen sets in soft topological spaces and study some of their properties.
Definition 10
A soft set (F,A) in a soft topological space X is called

(i)
soft bopen (sbopen) set iff (F,A)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))

(ii)
soft bclosed (sbclosed) set iff (F,A)\stackrel{~}{\supset}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cap}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))).
Theorem 1
For a soft set (F,A) in a soft topological space X

(i)
(F,A) is a soft bopen set iff {(F,A)}^{c} is a soft bclosed set.

(ii)
(F,A) is a soft bclosed set iff {(F,A)}^{c} is a soft bopen set.
Proof
Obvious from the Definition 10. \square
Definition 11
Let (X,\mathit{\tau},E) be a soft topological space and (F,A) be a soft set over X.

(i)
Soft bclosure of a soft set (F,A) in X is denoted by \mathrm{sbcl}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cap}\left\{(\mathrm{F},\mathrm{E})\stackrel{~}{\supset}(\mathrm{F},\mathrm{A}):(\mathrm{F},\mathrm{E})\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{0.277778em}{0ex}}\mathrm{a}\phantom{\rule{0.277778em}{0ex}}\mathrm{soft}\phantom{\rule{0.277778em}{0ex}}\mathrm{b}\text{}\mathrm{closed}\phantom{\rule{0.277778em}{0ex}}\mathrm{set}\phantom{\rule{0.277778em}{0ex}}\mathrm{of}\phantom{\rule{0.277778em}{0ex}}\mathrm{X}\right\}.

(ii)
Soft binterior of a soft set (F,A) in X is denoted by \mathrm{sbint}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cup}\left\{(\mathrm{O},\mathrm{A})\stackrel{~}{\subset}(\mathrm{F},\mathrm{A}):(\mathrm{O},\mathrm{A})\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{0.277778em}{0ex}}\mathrm{a}\phantom{\rule{0.277778em}{0ex}}\mathrm{soft}\phantom{\rule{0.277778em}{0ex}}\mathrm{b}\text{}\mathrm{open}\phantom{\rule{0.277778em}{0ex}}\mathrm{set}\phantom{\rule{0.277778em}{0ex}}\mathrm{of}\phantom{\rule{0.277778em}{0ex}}\mathrm{X}\right\}.
Clearly \mathrm{sbcl}((\mathrm{F},\mathrm{A})) is the smallest soft bclosed set over X which contains (F,A) and \mathrm{sbint}((\mathrm{F},\mathrm{A})) is the largest soft bopen set over X which is contained in (F,A).
Theorem 2
Let (F,A) be any soft set a in soft topological space X. Then,

(i)
\mathrm{sbcl}({(\mathrm{F},\mathrm{A})}^{\mathrm{c}})=\stackrel{~}{\mathrm{X}}\mathrm{sbint}((\mathrm{F},\mathrm{A})).

(ii)
\mathrm{sbint}({(\mathrm{F},\mathrm{A})}^{\mathrm{c}})=\stackrel{~}{\mathrm{X}}\mathrm{sbcl}((\mathrm{F},\mathrm{A})).
Proof
(i) Let sbopen set (O,A)\stackrel{~}{\subset}(F,A) and sbclosed set (F,E)\stackrel{~}{\supset}{(F,A)}^{c}. Then, \mathrm{sbint}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cup}\left\{{(\mathrm{F},\mathrm{E})}^{\mathrm{c}}:(\mathrm{F},\mathrm{E})\phantom{\rule{0.277778em}{0ex}}\text{is;a;softbclosed;set;and}\phantom{\rule{0.277778em}{0ex}}(\mathrm{F},\mathrm{E})\stackrel{~}{\supset}{(\mathrm{F},\mathrm{A})}^{\mathrm{c}}\right\}=\stackrel{~}{\mathrm{X}}\stackrel{~}{\cap}\{(\mathrm{F},\mathrm{E}):(\mathrm{F},\mathrm{E}) is a sbclosed set and (F,E)\stackrel{~}{\supset}{(F,A)}^{c}\}=\stackrel{~}{X}\mathrm{sbcl}({(\mathrm{F},\mathrm{A})}^{\mathrm{c}}).
Therefore, \mathrm{sbcl}({(\mathrm{F},\mathrm{A})}^{\mathrm{c}})=\stackrel{~}{\mathrm{X}}\mathrm{sbint}((\mathrm{F},\mathrm{A})).
(ii) Let (O,A) be sbopen set.
Then, for a sbclosed set (O,A)\stackrel{~}{\supset}(F,A), (O,A)\stackrel{~}{\subset}{(F,A)}^{c}.
Therefore, \mathrm{sbint}({(\mathrm{F},\mathrm{A})}^{\mathrm{c}})=\stackrel{~}{\mathrm{X}}\mathrm{sbcl}((\mathrm{F},\mathrm{A})). \square
Definition 12
Let (X,\mathit{\tau},E) be a soft topological space over X and (F,A) be a soft set over X.

(i)
[3] The soft interior of (F,A) is the soft set \mathrm{int}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cup}\left\{(\mathrm{O},\mathrm{A})\stackrel{~}{\subset}(\mathrm{F},\mathrm{A}):(\mathrm{O},\mathrm{A})\text{is;soft;open;of;}\mathrm{X}\right\};

(ii)
[2] The soft closure of (F,A) is the soft set \mathrm{cl}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cap}\left\{(\mathrm{F},\mathrm{E})\stackrel{~}{\supset}(\mathrm{F},\mathrm{A}):(\mathrm{F},\mathrm{E})\text{is;soft;closed;of;}\mathrm{X}\right\};

(iii)
[5] The soft semiinterior of (F,A) is a soft set \mathrm{ssint}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cup}\left\{(\mathrm{O},\mathrm{A})\stackrel{~}{\subset}(\mathrm{F},\mathrm{A}):(\mathrm{O},\mathrm{A})\phantom{\rule{0.333333em}{0ex}}\text{is;soft;semiopen;of;}\mathrm{X}\right\}.

(iv)
[5] The soft semiclosure of (F,A) is a soft set \mathrm{sscl}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cap}\{(\mathrm{F},\mathrm{E})\stackrel{~}{\supset}(\mathrm{F},\mathrm{A}):(\mathrm{F},\mathrm{E}) is soft semiclosed of X\};
The following concepts are used in the sequel.
Definition 13
A soft set (F,A) in a soft topological space X is called

(i)
soft regular open (soft regular closed) set [12] if (F,A)=\mathrm{cl}(\mathrm{int}(\mathrm{F},\mathrm{A})) (\mathrm{int}(\mathrm{cl}(\mathrm{F},\mathrm{A}))=(\mathrm{F},\mathrm{A})).

(ii)
soft \mathit{\alpha}open (soft \mathit{\alpha}closed) set [6] if (F,A)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}(\mathrm{int}(\mathrm{F},\mathrm{A})))(\mathrm{cl}(\mathrm{int}(\mathrm{cl}(\mathrm{F},\mathrm{A})))\stackrel{~}{\subset}(\mathrm{F},\mathrm{A})).

(iii)
soft preopen (soft preclosed) set [12] if (F,A)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}(\mathrm{F},\mathrm{A})) (\mathrm{cl}(\mathrm{int}(\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}(\mathrm{F},\mathrm{A})).

(iv)
soft semiopen (soft semiclosed) set [5] if (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}(\mathrm{F},\mathrm{A})) (\mathrm{int}(\mathrm{cl}(\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}(\mathrm{F},\mathrm{A})).

(v)
soft \mathit{\beta}open (soft \mathit{\beta}closed) set [12] if (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}(\mathrm{cl}(\mathrm{F},\mathrm{A})))(\mathrm{int}(\mathrm{cl}(\mathrm{int}(\mathrm{F},\mathrm{A})))\stackrel{~}{\subset}(\mathrm{F},\mathrm{A})).
Lemma 1
Let (F,A) be a soft set in a soft topological space X. Then

(i)
\mathrm{sscl}((\mathrm{F},\mathrm{A}))=(\mathrm{F},\mathrm{A})\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\phantom{\rule{1em}{0ex}}\mathrm{and}\mathrm{ssint}((\mathrm{F},\mathrm{A}))=(\mathrm{F},\mathrm{A})\stackrel{~}{\cap}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))).

(ii)
\mathrm{spcl}((\mathrm{F},\mathrm{A}))=(\mathrm{F},\mathrm{A})\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\phantom{\rule{1em}{0ex}}\mathrm{and}\mathrm{spint}((\mathrm{F},\mathrm{A}))=(\mathrm{F},\mathrm{A})\stackrel{~}{\cap}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))).
Proof
(i)
Also
Hence from (1) and (2), \mathrm{sscl}((\mathrm{F},\mathrm{A}))=(\mathrm{F},\mathrm{A})\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))).
(ii) Similar by taking the complements. \square
Lemma 2
In a soft topological space we have the following

(i)
Every soft regular open set is soft open.

(ii)
Every soft open set is soft \mathit{\alpha}open.

(iii)
Every soft \mathit{\alpha}open set is both soft semiopen and soft preopen.

(iv)
Every soft semiopen set and every soft preopen set is soft \mathit{\beta}open.
Let X be a soft topological space.Then, the family of all soft regular open (resp. soft \mathit{\alpha}open, soft semiopen, soft preopen, soft \mathit{\beta}open) sets in X may be denoted by sr (resp. saopen, ssopen, spopen, s\mathit{\beta}open) sets. The complement of the above represents the family of soft regular closed (resp. soft \mathit{\alpha}closed, soft semiclosed, soft preclosed, soft \mathit{\beta}closed) sets in X and may be denoted by sr (resp. s\mathit{\alpha}closed, ssclosed, spclosed, s\mathit{\beta}closed) sets. Also, the family of all soft bopen (resp. soft bclosed) sets in X may be denoted by \mathrm{SbOS}(\mathrm{X}) (resp. \mathrm{SbCS}(\mathrm{X})).
Theorem 3
In a soft topological space X

(i)
Every spopen set is sbopen set.

(ii)
Every ssopen set is sbopen set.
Proof
(i) Let (F,A) be a spopen set in a soft topological space X.
Then, (F,A)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))) which implies
(F,A)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{int}((\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))).
Thus (F,A) is sbopen set.
(ii) Let (F,A) be a ssopen set in a soft topological space X. Then, (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))) which implies (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{int}((\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))).
Thus (F,A) is sbopen set. \square
The converses are not true as seen in the following example:
Example 1
Let X=\{{h}_{1},{h}_{2,}{h}_{3,}{h}_{4}\}, E=\{{e}_{1},{e}_{2},{e}_{3}\} and
\mathit{\tau}=\{\mathrm{\varnothing},\stackrel{~}{X},({F}_{1},E),({F}_{2},E),({F}_{3},E),...,({F}_{15},E)\}, where ({F}_{1},E),({F}_{2},E),({F}_{3},E),...({F}_{15},E) are soft sets over X, defined as follows:
Then, \mathit{\tau} defines a soft topology on X, and thus (X,\mathit{\tau},E) is a soft topological space over X. Clearly, the soft closed sets are \stackrel{~}{X},\mathrm{\varnothing},{({F}_{1},E)}^{c},{({F}_{2},E)}^{c},{({F}_{3},E)}^{c},...,{({F}_{15},E)}^{c}.
Then, let us take (F,A)=\{({e}_{1},\{{h}_{2},{h}_{4}\}),({e}_{2},\{{h}_{1},{h}_{3}\}),({e}_{3},\{{h}_{1},{h}_{3},{h}_{4}\})\}; then \mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))=\stackrel{~}{\mathrm{X}}, and so (F,A)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))); hence, (F,A) is sbopen set, but not spopen set (since (F,A) is not spopen set).
Now, let us take (G,E)=\{({e}_{1},\{{h}_{4}\}),({e}_{2},\{{h}_{1},{h}_{2},{h}_{3}\}),({e}_{3},\{{h}_{2},{h}_{4}\})\}; then \mathrm{int}(\mathrm{cl}((\mathrm{G},\mathrm{E})))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{G},\mathrm{E})))=\stackrel{~}{\mathrm{X}}, and so (G,E)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}((\mathrm{G},\mathrm{E})))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{G},\mathrm{E}))); thus, (G,E) is sbopen set, but not ssopen set.
Remark 1

(i)
If (F,A) is a soft set of soft topological space X, then \mathrm{sbcl}((\mathrm{F},\mathrm{A})) is the smallest sbclosed set containing (F,A).
Thus, \mathrm{sbcl}(\mathrm{F},\mathrm{A})=(\mathrm{F},\mathrm{A})\stackrel{~}{\cup}[\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cap}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))].

(ii)
If (F,A) is a soft set of soft topological space X then \mathrm{sbint}((\mathrm{F},\mathrm{A})) is the largest sbopen set contained in (F,A).
Thus, \mathrm{sbint}(\mathrm{F},\mathrm{A})=(\mathrm{F},\mathrm{A})\stackrel{~}{\cap}[\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))].
Theorem 4
In a soft topological space X, every sbopen (sbclosed) set is s\mathit{\beta}open (s\mathit{\beta}closed) set.
Proof
Let (F,A) be a sbopen set in X.
Then (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))
\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))))\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))).
As a result (F,A) is s\mathit{\beta}open set. \square
The converse is not true as seen in the following example:
Example 2
Let X=\{{h}_{1},{h}_{2,}{h}_{3,}{h}_{4}\}, E=\{{e}_{1},{e}_{2},{e}_{3}\} and let (X,\mathit{\tau},E) be soft topological space over X. Let us consider the soft topology \mathit{\tau} on X given in Example 1; i.e., \mathit{\tau}=\{\mathrm{\varnothing},\stackrel{~}{X},({F}_{1},E),({F}_{2},E),({F}_{3},E),...,({F}_{15},E)\}.
Then, let us take (H,E)=\{({e}_{1},\{{h}_{1},{h}_{2}\}),({e}_{2},\{{h}_{3},{h}_{4}\}),({e}_{3},\{{h}_{1},{h}_{3},{h}_{4}\})\}; then \mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{H},\mathrm{E}))))=\{({\mathrm{e}}_{1},\stackrel{~}{\mathrm{X}}),({\mathrm{e}}_{2},\{{\mathrm{h}}_{1,}{\mathrm{h}}_{3},{\mathrm{h}}_{4}\}),({\mathrm{e}}_{3},\stackrel{~}{\mathrm{X}})\}, and so (H,E)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{H},\mathrm{E})))); therefore, (H,E) is s\mathit{\beta}open set but not sbopen set.
Remark 2
From the above theorems, we have the following
Theorem 5
In a soft topological space X

(i)
An arbitrary union of sbopen sets is a sbopen set.

(ii)
An arbitrary intersection of sbclosed sets is a sbclosed set.
Proof
(i) Let \{{(F,A)}_{\mathit{\alpha}}\} be a collection of sbopen sets. Then, for each \mathit{\alpha},
{(F,A)}_{\mathit{\alpha}}\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}({(\mathrm{F},\mathrm{A})}_{\mathit{\alpha}}))\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}({(\mathrm{F},\mathrm{A})}_{\mathit{\alpha}})). Now
Hence \stackrel{~}{\cup}({(F,A)}_{\mathit{\alpha}}) is a sbopen set.
(ii) Similarly by taking complements. \square
Theorem 6
In a soft topological space X, (F,A) is sbclosed (sbopen) set if and only if (F,A)=\mathrm{sbcl}(\mathrm{F},\mathrm{A}) ((F,A)=\mathrm{sbint}(\mathrm{F},\mathrm{A})).
Proof
Suppose (F,A)=\mathrm{sbcl}((\mathrm{F},\mathrm{A}))=\stackrel{~}{\cap}\left\{(F,E):(F,E)\phantom{\rule{0.333333em}{0ex}}\text{is;a;s}b\text{closed;set;and;}(F,E)\stackrel{~}{\supset}(F,A)\right\} that implies, (F,A)\in \stackrel{~}{\cap}\left\{(F,E)\phantom{\rule{0.333333em}{0ex}}\text{is;a;s}b\text{closed;set;and;}(F,E)\stackrel{~}{\supset}(F,A)\right\}, that implies (F,A) is sbclosed set.
Conversely, suppose (F,A) is a sbclosed set in X.
We take (F,A)\stackrel{~}{\subset}(F,A) and (F,A) is a sbclosed. Therefore,
(F,A)\in \stackrel{~}{\cap}\left\{(F,E):(F,E)\phantom{\rule{0.333333em}{0ex}}\text{is;a;s}\text{bclosed;set;and;}(F,E)\stackrel{~}{\supset}(F,A)\right\}.
(F,A)\stackrel{~}{\subset}(F,E) implies, (F,A)=\stackrel{~}{\cap}\{(F,E):(F,E) is a sbclosed set and (F,E)\stackrel{~}{\supset}(F,A)\}=\mathrm{sbcl}((\mathrm{F},\mathrm{A})).
For (F,A)=\mathrm{sbint}((\mathrm{F},\mathrm{A})) we apply soft interiors. \square
Theorem 7
In a soft topological space X the following hold for sbclosure.

(i)
\mathrm{sbcl}(\mathrm{\Phi})=\mathrm{\Phi}.

(ii)
\mathrm{sbint}(\mathrm{\Phi})=\mathrm{\Phi}.

(iii)
\mathrm{sbcl}(\mathrm{F},\mathrm{A}) is a sbclosed set in X.

(iv)
\mathrm{sbcl}(\mathrm{sbcl}(\mathrm{F},\mathrm{A}))=\mathrm{sbcl}((\mathrm{F},\mathrm{A})).
Proof
The proof is obvious. \square
Theorem 8
In a soft topological space X the following relations hold;

(i)
\mathrm{sbcl}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B}))\stackrel{~}{\supset}\mathrm{sbcl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cup}\mathrm{sbcl}((\mathrm{F},\mathrm{B})),

(ii)
\mathrm{sbcl}((\mathrm{F},\mathrm{A})\stackrel{~}{\cap}(\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}\mathrm{sbcl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cap}\mathrm{sbcl}((\mathrm{F},\mathrm{B})).
Proof
(i) (F,A)\stackrel{~}{\subset}(F,A)\stackrel{~}{\cup}(F,B) or (F,B)\stackrel{~}{\subset}(F,A)\stackrel{~}{\cup}(F,B) that implies \mathrm{sbcl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}\mathrm{sbcl}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B})) or \mathrm{sbcl}((\mathrm{F},\mathrm{B}))\stackrel{~}{\subset}\mathrm{sbcl}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B})).
Thus, \mathrm{sbcl}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B}))\stackrel{~}{\supset}\mathrm{sbcl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cup}\mathrm{sbcl}((\mathrm{F},\mathrm{B})). (ii) Similar to that of (i). \square
Theorem 9
In a soft topological space X the following relations hold;

(i)
\mathrm{sbint}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B}))\stackrel{~}{\supset}\mathrm{sbint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cup}\mathrm{sbint}((\mathrm{F},\mathrm{B})),

(ii)
\mathrm{sbint}((\mathrm{F},\mathrm{A})\stackrel{~}{\cap}(\mathrm{F},\mathrm{B}))\stackrel{~}{\subset}\mathrm{sbint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cap}\mathrm{sbint}((\mathrm{F},\mathrm{B})).
Proof
(i) (F,A)\stackrel{~}{\subset}(F,A)\stackrel{~}{\cup}(F,B) or (F,B)\stackrel{~}{\subset}(F,A)\stackrel{~}{\cup}(F,B) that implies \mathrm{sbint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}\mathrm{sbint}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B})) or \mathrm{sbint}((\mathrm{F},\mathrm{B}))\stackrel{~}{\subset}\mathrm{sbint}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B})).
Thus, \mathrm{sbint}((\mathrm{F},\mathrm{A})\stackrel{~}{\cup}(\mathrm{F},\mathrm{B}))\stackrel{~}{\supset}\mathrm{sbint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cup}\mathrm{sbint}((\mathrm{F},\mathrm{B})). (ii) Similar to that of (i). \square
Theorem 10
Let (F,A) be a sbopen set in a soft topological space X.

(i)
If (F,A) is a srclosed set then (F,A) is a spopen set.

(ii)
If (F,A) is a sropen set then (F,A) is a ssopen set.
Proof
Since (F,A) is sbopen set, (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))),
(i) Now let (F,A) be srclosed set. Therefore, (F,A)=\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))).
Then (F,A)\stackrel{~}{\subset}(F,A)\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))). That implies
(F,A)\stackrel{~}{\subset}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))).
Hence (F,A) is spopen set.
(ii) Let (F,A) be sropen set. Therefore, (F,A)=\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))).
Then (F,A)\stackrel{~}{\subset}(F,A)\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))).
That implies (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))). Thus (F,A) is ssopen set. \square
Theorem 11
Let (F,A) be a sbopen set in a soft topological space X.

(i)
If (F,A) is a srclosed set then (F,A) is a ssclosed set.

(ii)
If (F,A) is a sropen set then (F,A) is a spclosed set.
Theorem 12
For any sbopen set (F,A) in soft topological space X, \mathrm{cl}((\mathrm{F},\mathrm{A})) is srclosed set.
Proof
Let (F,A) be sbopen set in X which implies (F,A)\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))). Therefore, \mathrm{cl}(\mathrm{F},\mathrm{A})\stackrel{~}{\subset}\mathrm{cl}(\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A}))))\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))))\stackrel{~}{\subset}\mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))).
Also \mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))))\stackrel{~}{\subset}\mathrm{cl}((\mathrm{F},\mathrm{A})). Thus \mathrm{cl}(\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A}))))=\mathrm{cl}((\mathrm{F},\mathrm{A})).
So \mathrm{cl}((\mathrm{F},\mathrm{A})) is srclosed set. \square
Theorem 13
For any sbclosed set (F,A) in a soft topological space X.\mathrm{int}((\mathrm{F},\mathrm{A})) is sropen set.
Proof
Let (F,A) be sbclosed set in X which implies \mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cap}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\subset}(\mathrm{F},\mathrm{A}). Thus
Also
Thus from (3) and (4) \mathrm{int}((\mathrm{F},\mathrm{A}))=\mathrm{int}(\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))). So \mathrm{int}((\mathrm{F},\mathrm{A})) is sropen set. \square
Theorem 14
Let (F,A) be a sbopen (sbclosed) set in a soft topological space X, such that \mathrm{int}((\mathrm{F},\mathrm{A}))=\mathrm{\Phi}. Then, (F,A) is a spopen set.
Theorem 15
Let (F,A) be a sbopen (sbclosed) set in a soft topological space X, such that \mathrm{cl}((\mathrm{F},\mathrm{A}))=\mathrm{\Phi}. Then, (F,A) is a ssopen set.
Theorem 16
Let (F,A) be a set of a soft topological space X. Then,

(i)
\mathrm{sbcl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}\mathrm{sscl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cap}\mathrm{spcl}((\mathrm{F},\mathrm{A})).

(ii)
\mathrm{sbint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\supset}\mathrm{ssint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cup}\mathrm{spint}((\mathrm{F},\mathrm{A})).
Proof
(i) \mathrm{sscl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cap}\mathrm{spcl}((\mathrm{F},\mathrm{A}))\stackrel{~}{\supset}[(F,A)\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))]\stackrel{~}{\cap}[(\mathrm{F},\mathrm{A})\stackrel{~}{\cup}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))]=(F,A)\stackrel{~}{\cup}[\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cap}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))]=\mathrm{sbcl}((\mathrm{F},\mathrm{A})).
(ii) \mathrm{ssint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\cup}\mathrm{spint}((\mathrm{F},\mathrm{A}))\stackrel{~}{\subset}[(F,A)\stackrel{~}{\cap}\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))]\stackrel{~}{\cup}[(\mathrm{F},\mathrm{A})\stackrel{~}{\cap}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))]=(F,A)\stackrel{~}{\cap}[\mathrm{cl}(\mathrm{int}((\mathrm{F},\mathrm{A})))\stackrel{~}{\cup}\mathrm{int}(\mathrm{cl}((\mathrm{F},\mathrm{A})))]=\mathrm{sbint}((\mathrm{F},\mathrm{A})). \square
Theorem 17
Let X be a soft topological space. If (F,B) is an soft open set and (F,A) is a sbopen set in X. Then (F,A)\stackrel{~}{\cap}(F,B) is a sbopen set in X.
Proof
Hence (F,A)\stackrel{~}{\cap}(F,B) is a sbopen set in X. \square
Theorem 18
Let X be soft topological space. If (F,B) is an s\mathit{\alpha}open set and (F,A) is a sbopen set in X. Then (F,A)\stackrel{~}{\cap}(F,B) is a sbopen set in X.
Proof
Hence (F,A)\stackrel{~}{\cap}(F,B) is sbopen set in X. \square
Theorem 19
(B,E) is sbopen (sbclosed) set in X if and only if (B,E) is the union (intersection) of ssopen set and sbopen set in X.
Proof
Follows from the Definitions 11. \square
Soft bcontinuity
In this section, we introduce soft bcontinuous maps, soft birresolute maps, soft bclosed maps and soft bopen maps and study some of their properties.
Definition 14
([13]) let (X,E) and (Y,K) be soft classes. Let u:X\to Y and p:E\to K be mappings. Then a mapping f:(X,E)\to (Y,K) is defined as: for a soft set (F,A) in (X,E), (f\left(F,A),B\right),B=p(A)\subseteq K is a soft set in (Y,K) given by f\left(F,A\right)\left(\mathit{\beta}\right)=u\left(\underset{\mathit{\alpha}\in {p}^{1}\left(\mathit{\beta}\right)\cap A}{\cup F\left(\mathit{\alpha}\right)}\right) for \mathit{\beta}\in K.(f\left(F,A),B\right) is called a soft image of a soft set (F,A). If B=K, then we shall write (f\left(F,A),K\right) as f(F,A).
Definition 15
[13] Let f:(X,E)\to (Y,K) be a mapping from a soft class (X,E) to another soft class (Y,K), and (G,C) a soft set in soft class (Y,K), where C\subseteq K. Let u:X\to Y and p:E\to K be mappings. Then \left({f}^{1}\left(G,C\right),D\right),D={p}^{1}\left(C\right), is a soft set in the soft classes (X,E), defined as: {f}^{1}\left(G,C\right)\left(\mathit{\alpha}\right)={u}^{1}\left(G\left(p\left(\mathit{\alpha}\right)\right)\right) for \mathit{\alpha}\in D\subseteq E.\left({f}^{1}\left(G,C\right),D\right) is called a soft inverse image of \left(G,C\right). Hereafter, we shall write \left({f}^{1}\left(G,C\right),E\right) as {f}^{1}\left(G,C\right).
Theorem 20
[13] Let f:(X,E)\to (Y,K),u:X\to Y and p:E\to K be mappings. Then for soft sets \left(F,A\right),\left(G,B\right) and a family of soft sets \left({F}_{i},{A}_{i}\right) in the soft class \left(X,E\right), we have:

(1)
f\left(\mathrm{\Phi}\right)=\mathrm{\Phi},

(2)
f\left(\stackrel{\sim}{X}\right)=\stackrel{\sim}{Y},

(3)
f\left((F,A)\stackrel{\sim}{\cup}\left(G,B\right)\right)=f(F,A)\stackrel{\sim}{\cup}f\left(G,B\right) in general f\left({\cup}_{i}\left({F}_{i},{A}_{i}\right)\right)={\cup}_{i}f\left({F}_{i},{A}_{i}\right),

(4)
f\left((F,A)\stackrel{\sim}{\cap}\left(G,B\right)\right)\stackrel{\sim}{\subseteq}f(F,A)\stackrel{\sim}{\cap}f\left(G,B\right) in general f\left({\cap}_{i}\left({F}_{i},{A}_{i}\right)\right)\stackrel{\sim}{\subseteq}{\cap}_{i}f\left({F}_{i},{A}_{i}\right),

(5)
If (F,A)\stackrel{\sim}{\subseteq}\left(G,B\right), then f(F,A)\stackrel{\sim}{\subseteq}f\left(G,B\right),

(6)
{f}^{1}\left(\mathrm{\Phi}\right)=\mathrm{\Phi},

(7)
{f}^{1}\left(\stackrel{\sim}{Y}\right)=\stackrel{\sim}{X},

(8)
{f}^{1}\left((F,A)\stackrel{\sim}{\cup}\left(G,B\right)\right)={f}^{1}(F,A)\stackrel{\sim}{\cup}{f}^{1}\left(G,B\right) in general {f}^{1}\left({\stackrel{~}{\cup}}_{i}\left({F}_{i},{A}_{i}\right)\right)={\stackrel{~}{\cup}}_{i}{f}^{1}\left({F}_{i},{A}_{i}\right),

(9)
{f}^{1}\left((F,A)\stackrel{\sim}{\cap}\left(G,B\right)\right)={f}^{1}(F,A)\stackrel{\sim}{\cap}{f}^{1}\left(G,B\right) in general {f}^{1}\left({\stackrel{~}{\cap}}_{i}\left({F}_{i},{A}_{i}\right)\right)={\stackrel{~}{\cap}}_{i}{f}^{1}\left({F}_{i},{A}_{i}\right),

(10)
If (F,A)\stackrel{\sim}{\subseteq}\left(G,B\right), then {f}^{1}(F,A)\stackrel{\sim}{\subseteq}{f}^{1}\left(G,B\right).
Definition 16
A soft mapping f:X\to Y is said to be soft bcontinuous (briefly sbcontinuous) if the inverse image of each soft open set of Y is a sbopen set in X.
Theorem 21
Let f:X\to Y be a mapping from a soft space X to soft space Y. Then, the following statements are true;

(i)
f is sbcontinuous,

(ii)
the inverse image of each soft closed set in Y is soft bclosed in X.
Proof
(i)\Rightarrow(ii): Let (G,K) be a soft closed set in Y. Then {(G,K)}^{c} is soft open set. Thus, {f}^{1}({(G,K)}^{c})\in \mathrm{SbOS}(\mathrm{X}), i.e., X{f}^{1}((G,K))\in \mathrm{SbOS}(\mathrm{X}). Hence {f}^{1}((G,K)) is a sbclosed set in X.
(ii)\Rightarrow(i): Let (O,K) is soft open set in Y. Then {(O,K)}^{c} is soft closed set and by (ii) we have {f}^{1}({(O,K)}^{c})\in \mathrm{SbCS}(\mathrm{X}), i.e., X{f}^{1}((O,K))\in \mathrm{SbCS}(\mathrm{X}). Hence {f}^{1}((O,K)) is a sbopen set in X. Therefore, f is a sbcontinuous function. \square
Definition 17
A soft mapping f:X\to Y is called soft \mathit{\beta}continuous [9] (resp. soft \mathit{\alpha}continuous [6], soft precontinuous [6], soft semicontinuous [8]) if the inverse image of each soft open set in Y is s\mathit{\beta}open (resp. s\mathit{\alpha}open, spopen, ssopen) set in X.
Remark 3
We have following implications, however, examples given below show that the converses of these implications are not true.
Example 3
f is a soft \mathit{\alpha}continuous function, but not soft continuous function given Example 4 in [6].
Example 4
f is a soft precontinuous function and soft, but not soft \mathit{\alpha}continuous function given Example 5 in [6].
Example 5
f is a soft semicontinuous function and soft, but not soft \mathit{\alpha}continuous function given Example 6 in [6].
Example 6
Let X=\{{h}_{1},{h}_{2,}{h}_{3},{h}_{4}\}, Y=\{{y}_{1},{y}_{2},{y}_{3},{y}_{4}\}, E=\{{e}_{1},{e}_{2},{e}_{3}\}, and K=\{{k}_{1},{k}_{2},{k}_{3}\} and let (X,\mathit{\tau},E) and (Y,\mathit{\upsilon},K) be soft topological spaces.
Define u:X\to Y and p:E\to K as
u({h}_{1})=\{{y}_{1}\}, u({h}_{2})=\{{y}_{3}\}, u({h}_{3})=\{{y}_{2}\}, u({h}_{4})=\{{y}_{4}\},
p({e}_{1})=\{{k}_{2}\}, p({e}_{2})=\{{k}_{1}\}, p({e}_{3})=\{{k}_{3}\}.
Let us consider the soft topology \mathit{\tau} on X given in Example 1; i.e.,
\mathit{\tau}=\{\mathrm{\Phi},\stackrel{\sim}{X},({F}_{1},E),({F}_{2},E),({F}_{3},E),...({F}_{15},E)\}, \mathit{\upsilon}=\{\mathrm{\Phi},\stackrel{\sim}{Y},(L,K)\};
(L,K)=\{({k}_{1},\{{y}_{2},{y}_{4}\}),({k}_{2},\{{y}_{1},{y}_{3}\}),({k}_{3},\{{y}_{1},{y}_{2},{y}_{4}\})\} and mapping;
f:(X,\mathit{\tau},E)\to (Y,\mathit{\upsilon},K) is a soft mapping. Then (L,K) is a soft open set in Y;{f}^{1}((L,K))=\{({e}_{1},\{{h}_{1},{h}_{2}\}),({e}_{2},\{{h}_{3},{h}_{4}\}),({e}_{3},\{{h}_{1},{h}_{3},{h}_{4}\})\} is a s\mathit{\beta}open set but not sbopen set in X. Therefore, f is a soft \mathit{\beta}continuous function but not sbcontinuous function.
Example 7
Let X=\{{h}_{1},{h}_{2,}{h}_{3},{h}_{4}\}, Y=\{{y}_{1},{y}_{2},{y}_{3},{y}_{4}\}, E=\{{e}_{1},{e}_{2},{e}_{3}\}, and K=\{{k}_{1},{k}_{2},{k}_{3}\} and let (X,\mathit{\tau},E) and (Y,\mathit{\upsilon},K) be soft topological spaces.
Define u:X\to Y and p:E\to K as
u({h}_{1})=\{{y}_{1}\}, u({h}_{2})=\{{y}_{3}\}, u({h}_{3})=\{{y}_{2}\}, u({h}_{4})=\{{y}_{4}\},
p({e}_{1})=\{{k}_{2}\}, p({e}_{2})=\{{k}_{1}\}, p({e}_{3})=\{{k}_{3}\}.
Let us consider the soft topology \mathit{\tau} on X given in Example 1; i.e.,
\mathit{\tau}=\{\mathrm{\Phi},\stackrel{\sim}{X},({F}_{1},E),({F}_{2},E),({F}_{3},E),...({F}_{15},E)\}, \mathit{\upsilon}=\{\mathrm{\Phi},\stackrel{\sim}{Y},(M,K)\};
(M,K)=\{({k}_{1},\{{y}_{1},{y}_{2},{y}_{3}\}),({k}_{2},\{{y}_{4}\}),({k}_{3},\{{y}_{3},{y}_{4}\})\} and mapping;
f:(X,\mathit{\tau},E)\to (Y,\mathit{\upsilon},K) is a soft mapping. Then (M,K) is a soft open set in Y; {f}^{1}((M,K))=\{({e}_{1},\{{h}_{4}\}),({e}_{2},\{{h}_{1},{h}_{2},{h}_{3}\}),({e}_{3},\{{h}_{2},{h}_{4}\})\} is a sbopen set but not ssopen set in X. Hence, f is a sbcontinuous function but not soft semicontinuous function.
Example 8
Let X=\{{h}_{1},{h}_{2,}{h}_{3},{h}_{4}\}, Y=\{{y}_{1},{y}_{2},{y}_{3},{y}_{4}\}, E=\{{e}_{1},{e}_{2},{e}_{3}\}, and K=\{{k}_{1},{k}_{2},{k}_{3}\} and let (X,\mathit{\tau},E) and (Y,\mathit{\upsilon},K) be soft topological spaces.
Define u:X\to Y and p:E\to K as
u({h}_{1})=\{{y}_{1}\}, u({h}_{2})=\{{y}_{3}\}, u({h}_{3})=\{{y}_{2}\}, u({h}_{4})=\{{y}_{4}\},
p({e}_{1})=\{{k}_{2}\}, p({e}_{2})=\{{k}_{1}\}, p({e}_{3})=\{{k}_{3}\}.
Let us consider the soft topology \mathit{\tau} on X given in Example 1; i.e.,
\mathit{\tau}=\{\mathrm{\Phi},\stackrel{\sim}{X},({F}_{1},E),({F}_{2},E),({F}_{3},E),...({F}_{15},E)\}, \mathit{\upsilon}=\{\mathrm{\Phi},\stackrel{\sim}{Y},(N,K)\};
(N,K)=\{({k}_{1},\{{y}_{1},{y}_{2}\}),({k}_{2},\{{y}_{3},{y}_{4}\}),({k}_{3},\{{y}_{1},{y}_{2},{y}_{4}\})\} and mapping;
f:(X,\mathit{\tau},E)\to (Y,\mathit{\upsilon},K) is a soft mapping. Then (N,K) is a soft open set in Y; {f}^{1}((N,K))=\{({e}_{1},\{{h}_{2},{h}_{4}\}),({e}_{2},\{{h}_{1},{h}_{3}\}),({e}_{3},\{{h}_{1},{h}_{3},{h}_{4}\})\} is a sbopen set but not spopen set in X. Thus, f is a sbcontinuous function, but not soft precontinuous function.
Theorem 22
Every soft continuous function is sbcontinuous function.
Proof
Let f:X\to Y be a soft continuous function. Let (F,K) be a soft open set in Y. Since f is soft continuous, {f}^{1}((F,K)) is soft open in X. And so {f}^{1}((F,K)) is sbopen set in X. Therefore, f is sbcontinuous function. \square
Definition 18
A mapping f:X\to Y is said to be soft birresolute (briefly sbirresolute) if {f}^{1}((F,K)) is sbclosed set in X, for every sbclosed set (F,K) in Y.
Theorem 23
A mapping f:X\to Y is sbirresolute mapping if and only if the inverse image of every sbopen set in Y is sbopen set in X.
Theorem 24
Every sbirresolute mapping is sbcontinuous mapping.
Proof
Let f:X\to Y is sbirresolute mapping. Let (F,K) be a soft closed set in Y, then (F,K) is sbclosed set in Y. Since f is sbirresolute mapping, {f}^{1}((F,K)) is a sbclosed set in X. Hence, f is sbcontinuous mapping. \square
Theorem 25
Let f:(X,\mathit{\tau},E)\to (Y,v,K), g:(Y,\mathit{\upsilon},K)\to (Z,\mathit{\sigma},T) be two functions. Then

(i)
g\circ f:X\to Z is sbcontinuous, if f is sbcontinuous and g is soft continuous.

(ii)
g\circ f:X\to Z is sbirresolute, if f and g is sbirresolute functions.

(iii)
g\circ f:X\to Z is sbcontinuous if f is sbirresolute and g is sbcontinuous.
Proof

(i)
Let (H,T) be soft closed set of Z. Since g:Y,\to Z is soft continuous, by definition {g}^{1}((H,T)) is soft closed set of Y. Now f:X\to Y is sbcontinuous and {g}^{1}((H,T)) is soft closed set of Y, so by Definition 16, {f}^{1}({g}^{1}((H,T)))={(g\circ f)}^{1}((H,T)) is sbclosed in X. Hence g\circ \phantom{\rule{4pt}{0ex}}f:X\to Z is sbcontinuous.

(ii)
Let g:Y\to Z is sbirresolute and let (H,T) be sbclosed set of Z. Since g is sbirresolute by Definition 18, {g}^{1}((H,T)) is sbclosed set of Y. Also f:X\to Y is sbirresolute, so {f}^{1}({g}^{1}((H,T)))={(g\circ f)}^{1}((H,T)) is sbclosed. Thus, g\circ \phantom{\rule{4pt}{0ex}}f:X\to Z is sbirresolute.

(iii)
Let (H,T) be sbclosed set of Z. Since g:Y\to Z is sbcontinuous, {g}^{1}((H,T)) is sbclosed set of Y. Also f:X\to Y is sbirresolute, so every sbclosed set of Y is sbclosed in X. Therefore, {f}^{1}({g}^{1}((H,T)))={(g\circ f)}^{1}((H,T)) is sbclosed set of X. Thus, g\circ f:X\to Z is sbcontinuous.
\square
Theorem 26
If the bijective map f:X\to Y is soft open and sbirresolute, then f is sbirresolute.
Proof
Let (F,K) be a sbclosed set in Y and let {f}^{1}((F,K))\stackrel{~}{\subset}(F,A) where (F,A) is a soft open set in X. Clearly, (F,K)\stackrel{~}{\subset}f(F,A). Since f:X\to Y is soft open map, by definition f((F,A)) is soft open in Y and (F,K) is sbclosed set in Y. Then \mathrm{sbcl}((\mathrm{F},\mathrm{K}))\stackrel{~}{\subset}\mathrm{f}((\mathrm{F},\mathrm{A})), and hence {f}^{1}(\mathrm{sbcl}((\mathrm{F},\mathrm{K})))\stackrel{~}{\subset}(\mathrm{F},\mathrm{A}). Also f is sbirresolute map and \mathrm{sbcl}((\mathrm{F},\mathrm{K})) is a sbclosed set in Y, then {f}^{1}(\mathrm{sbcl}((\mathrm{F},\mathrm{K}))) is sbclosed set in X. Thus, \mathrm{sbcl}({\mathrm{f}}^{1}((\mathrm{F},\mathrm{K})))\stackrel{~}{\subset}\mathrm{sbcl}({\mathrm{f}}^{1}(\mathrm{sbcl}((\mathrm{F},\mathrm{K}))))\stackrel{~}{\subset}(\mathrm{F},\mathrm{A}). So {f}^{1}((F,K)) is sbclosed set in X. Therefore, f:X\to Y is sbirresolute map. \square
Definition 19
A mapping f:X\to Y is said to be soft bopen (briefly sbopen) map if the image of every soft open set in X is sbopen set in Y.
Definition 20
A mapping f:X\to Y is said to be soft bclosed (briefly sbclosed) map if the image of every soft closed set in X is sbclosed set in Y.
Theorem 27
If f:X\to Y is soft closed function and g:Y\to Z is sbclosed function, then g\circ f is sbclosed function.
Proof
For a soft closed set (F,A) in X, f((F,A)) is soft closed set in Y. Since g:Y\to Z is sbclosed function, g(f((F,A))) is sbclosed set in Z. g(f((F,A)))=(g\circ f)((F,A)) is sbclosed set in Z. Therefore, g\circ f is sbclosed function. \square
Theorem 28
A map f:X\to Y is sbclosed if and only if for each soft set (F,K) of Y and for each soft open set (F,A) such that {f}^{1}((F,K))\stackrel{~}{\subset}(F,A), there is a sbopen set (G,K) of Y such that (F,K)\stackrel{~}{\subset}(G,K) and {f}^{1}(G,K)\stackrel{~}{\subset}(F,A).
Proof
Suppose f is sbclosed map. Let (F,K) be a soft set of Y, and (F,A) be a soft open set of X, such that {f}^{1}((F,K))\stackrel{~}{\subset}(F,A). Then (G,K)={(f({(F,A)}^{c}))}^{c} is a sbopen set in Y such that (F,K)\stackrel{~}{\subset}(G,K) and {f}^{1}((G,K))\stackrel{~}{\subset}(F,A).
Conversely, suppose that (F,B) is a soft closed set of X. Then {f}^{1}({(f((F,B)))}^{c})\stackrel{~}{\subset}{(F,B)}^{c}, and {(F,B)}^{c} is soft open set. By hypothesis, there is a sbopen set (G,K) of Y such that (f{({(F,A)}^{c})}^{c}\stackrel{~}{\subset}(G,K) and {f}^{1}((G,K))\stackrel{~}{\subset}(F,B). Thus (F,B)\stackrel{~}{\subset}{f}^{1}((G,K)). Hence {(G,K)}^{c}\stackrel{~}{\subset}f((G,K))\stackrel{~}{\subset}f({({f}^{1}((G,K)))}^{c})\stackrel{~}{\subset}(G,K), which implies f((F,B))={(G,K)}^{c}. Since {(G,K)}^{c} is sbclosed set, f((F,B)) is sbclosed set. So f is a sbclosed map. \square
Theorem 29
Let f:X\to Y, g:Y\to Z be two maps such that g\circ f:X\to Z is sbclosed map.

(i)
If f is soft continuous and surjective, then g is sbclosed map.

(ii)
If g is sbirresolute and injective, then f is sbclosed map.
Proof

(i)
Let (H,K) be a soft closed set of Y. Then, {f}^{1}((H,K)) is soft closed set in X as f is soft continuous. Since g\circ f is sbclosed map, (g\circ f)({f}^{1}((H,K)))=g((H,K)) is sbclosed set in Z. Hence g:Y\to Z sbclosed map.

(ii)
Let (H,E) be a soft closed set in X. Then, (g\circ f)((H,E)) is sbclosed set in Z, and so {g}^{1}(g\circ f)((H,E))=f((H,E)) is sbclosed set in Y. Since g is sbirresolute and injective. Hence f is a sbclosed map.
\square
Theorem 30
If (F,B) is sbclosed set in X and f:X\to Y is bijective, soft continuous and sbclosed, then f((F,B)) is sbclosed set in Y.
Proof
Let f((F,B))\stackrel{~}{\subset}(F,K) where (F,K) is a soft open set in Y.
Since f is soft continuous, {f}^{1}((F,K)) is a soft open set containing (F,B).
Hence \mathrm{sbcl}((\mathrm{F},\mathrm{B}))\stackrel{~}{\subset}{\mathrm{f}}^{1}((\mathrm{F},\mathrm{K})) as (F,B) is sbclosed set.
Since f is sbclosed, f(\mathrm{sbcl}((\mathrm{F},\mathrm{B}))) is sbclosed set contained in the soft open set (F,K),
which implies \mathrm{sbcl}(\mathrm{f}(\mathrm{sbcl}((\mathrm{F},\mathrm{B}))))\stackrel{~}{\subset}(\mathrm{F},\mathrm{K}) and hence \mathrm{sbcl}(\mathrm{f}((\mathrm{F},\mathrm{B})))\stackrel{~}{\subset}(\mathrm{F},\mathrm{K}). So f((F,B)) is sbclosed set in Y. \square
Conclusion
In this paper, we introduce the concept of soft bopen sets and soft bcontinuous functions in topological spaces and some of their properties are studied. We also introduce soft binterior and soft bclosure and have established several interesting properties. In the end, we hope that this paper is just a beginning of a new structure, it will be necessary to carry out more theoretical research to promote a general framework for the practical application.
References
Molodtsov, D.: Soft set theoryFirst results. Comput. Math. Appl. 37(4–5), 19–31 (1999)
Shabir, M., Naz, M.: On soft topological spaces. Comput. Math. Appl. 61, 1786–1799 (2011)
Zorlutuna, I., Akdag, M., Min, W.K., Atmaca, S.: Remarks on soft topological spaces. Ann. Fuzzy Math. Inf. 3(2), 171–185 (2012)
Aygunoglu, A., Aygun, H: Some notes on soft topological spaces. Neural Comput. Appl. doi:10.1007/s0052101107223
Chen, B.: Soft semiopen sets and related properties in soft topological spaces. Appl. Math. Inf. Sci. 7(1), 287–294 (2013)
Akdag, M., Ozkan, A: Soft \mathit{\alpha}open sets and soft \mathit{\alpha}continuous functions. Abstr. Anal. Appl. Art ID 891341, 1–7 (2014)
Aras, C.G., Sonmez, A., Cakallı, H: On soft mappings. http://arxiv.org/abs/1305.4545 (Submitted)
Mahanta, J., Das, PK: On soft topological space via semiopen and semiclosed soft sets. http://arxiv.org/abs/1203.4133 (Submitted)
Yumak, Y., Kaymakcı, AK: Soft \mathit{\beta}open sets and their aplications. http://arxiv.org/abs/1312.6964 (Submitted)
Maji, P.K., Biswas, R., Roy, A.R.: Soft set theory. Comput. Math. Appl. 45, 555–562 (2003)
Ali, M.I., Feng, F., Liu, X., Min, W.K., Shabir, M.: On some new operations in soft set theory. Comput. Math. Appl. 57, 1547–1553 (2009)
Arockiarani, I., Arokialancy, A.: Generalized soft g\mathit{\beta}closed sets and soft gs\mathit{\beta}closed sets in soft topological spaces. Int. J. Math. Arch. 4(2), 1–7 (2013)
Kharal, A., Ahmad, B.: Mappings on soft classes. New Math. Nat. Comput. 7(3), 471–481 (2011)
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Akdag, M., Ozkan, A. Soft bopen sets and soft bcontinuous functions. Math Sci 8, 124 (2014). https://doi.org/10.1007/s4009601401247
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DOI: https://doi.org/10.1007/s4009601401247
Keywords
 Soft bopen (sbopen) sets
 Soft bclosed (sbclosed) sets
 Soft bcontinuous functions
Mathematics Subject Classification (2010)
 Primary 54C08
 Secondary 54C10