EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 4, 2017, 835-849
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On new classes of soft sets and functions via supra pre
open soft sets

F. A. Gharib1,∗, M. Ezzat Mohamed1,2, A. I. El-Maghrabi3,4

1 Faculty of Arts and Science, Northern Border University, Rafha, K. S. A.
2 Mathematics Department, Faculty of Science, Fayoum University,Fayoum, Egypt
3Department, Faculty of Science, Yanbu, Branch of Tibah University, Yanbu Al-Baher, K.
S. A.
4 Mathematics Department, Faculty of Science, Kafr El-Sheikh University, Kafr El-Sheikh,
Egypt.

Abstract. In this paper, we generalize the notions of supra soft locally closed sets [1] and supra
soft α-locally closed sets [2] by using the notions of supra pre-open soft sets [13]. Especially, we
introduce the notions of supra soft P -locally closed sets, supra soft P ∗-locally closed sets and supra
soft P ∗∗-locally closed sets in supra soft topological spaces. Also, we discuss their relationships
with other supra open soft sets in detail, supported by examples and counterexamples. These
examples illustrating the notions used in the paper are included. So we can see that all these
concepts are independent from each other or does implies the other. Also, we introduce three
different notions of generalized supra soft continuity, namely supra SPLC-continuous functions,
supra SP ∗LC-continuous functions and supra SP ∗∗LC-continuous functions. Furthermore, we
investigated some relations of these functions with other types of soft functions.

2010 Mathematics Subject Classifications: 54A05, 54A40, 54B05, 06D72

Key Words and Phrases: Supra soft topological space, Supra A-soft sets, Supra soft P -locally
closed sets, Supra SPLC-continuous functions.

1. Introduction

In 1983, Mashhour et al. [17] introduced the supra topological spaces, not only, as a
generalization to the class of topological spaces, but also, these spaces were easier in the
application as shown in [11]. In 2001, Popa et al. [19] generalized the supra topological
spaces to the minimal spaces and generalized spaces as a new wider classes. In 2007, Arpad
Szaz [12] succeed to introduce an application on the minimal spaces and generalized spaces.
In 1987, Abd El-Monsef et al. [9] introduced the fuzzy supra topological spaces. In 2001,

∗Corresponding author.

Email addresses: fatouhalmg@yahoo.com (F. A. Gharib), mohaezzat@yahoo.com (M. Ezzat Mohamed),
aelmaghrabi@yahoo.com (A. I. El-Maghrabi)

http://www.ejpam.com 835 c© 2017 EJPAM All rights reserved.



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 836

El-Sheikh success to use the fuzzy supra topology to study some topological properties to
the fuzzy bitopological spaces.

The notions of supra soft topological space were first introduced by El-Sheikh et al.
[13]. Various applications and topological properties on supra soft topological spaces were
introduced recently in [[3-8], [16]]. Properties of soft category and homotopy are introduced
in [22,23,24].

A. M. Abd El-latif [1], introduced the concepts of supra soft locally closed sets and
supra SLC-continuous functions in supra soft topological spaces. Our aim of this paper,
is to generalize these notions by using the notion of supra pre-open soft sets and discuss
some of their basic properties.

2. Preliminaries

In this section, we present the basic definitions and results of soft set theory and supra
soft topology.

Definition 1 (18). 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 non-empty subset of E. A pair (F,A) denoted by
FA is called a soft set over X , where F is a mapping given by F : A → P (X). In other
words, a soft set over X is a parametrized family of subsets of the universe X. For a
particular e ∈ A , F (e) may be considered the set of e-approximate elements of the soft
set (F,A) and if e 6∈ A, then F (e) = ϕ i.e
(F,A) = {(e, F (e)) : e ∈ A ⊆ E, F : A→ P (X)}. The family of all these soft sets denoted
by SS(X)A.

Definition 2 (21). Let τ be a collection of soft sets over a universe X with a fixed set of
parameters E, then τ ⊆ SS(X)E is called a soft topology on X if

(1) X̃, ϕ̃ ∈ τ , where ϕ̃(e) = ϕ and X̃(e) = X, ∀e ∈ E,

(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 3 (26). The soft set (F,E) ∈ SS(X)E is called a soft point in X̃ if there exist
x ∈ X and e ∈ E such that F (e) = {x} and F (ec) = ϕ for each ec ∈ E−{e}, and the soft
point (F,E) is denoted by xe.
The soft point xe is said to be belonging to the soft set (G,E), denoted by xe∈̃(G,E), if
for the element e ∈ E, F (e) ⊆ G(e).

Definition 4 (13). Let τ be a collection of soft sets over a universe X with a fixed set of
parameters E, then µ ⊆ SS(X)E is called supra soft topology on X with a fixed set E if

(1) X̃, ϕ̃ ∈ µ,



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 837

(2) the union of any number of soft sets in µ belongs to µ.

The triplet (X,µ,E) is called supra soft topological space (or supra soft spaces) over
X.

Definition 5 (13). Let (X, τ,E) be a soft topological space and (X,µ,E) be a supra soft
topological space. We say that, µ is a supra soft topology associated with τ if τ ⊂ µ.

Definition 6 (13). Let (X,µ,E) be a supra soft topological space over X, then the members
of µ are said to be supra open soft sets in X. We denote the set of all supra open soft sets
over X by supra-OS(X,µ,E), or when there can be no confusion by supra-OS(X) and the
set of all supra closed soft sets by supra-CS(X,µ,E), or supra-CS(X).

Definition 7 (13). Let (X,µ,E) be a supra soft topological space over and (F,E) ∈
SS(X)E. Then, the supra soft interior of (G,E), denoted by ints(G,E) is the soft union
of all supra open soft subsets of (G,E).i.e
ints(G,E) = ∪̃{(H,E) : (H,E) is supra open soft set and (H,E)⊆̃(G,E)}.
Also, the supra soft closure of (F,E), denoted by cls(F,E) is the soft intersection of all
supra closed super soft sets of (F,E) i.e
cls(F,E) = ∩̃{(H,E) : (H,E) is supra closed soft set and (F,E)⊆̃(H,E)}.

Definition 8 (1,3,13). Let (X,µ,E) be a supra soft topological space and (F,E) ∈ SS(X)E.
Then, (F,E) is said to be,

(1) Supra pre open soft set if (F,E)⊆̃ints(cls(F,E)).

(2) Supra semi open soft set if (F,E)⊆̃cls(ints(F,E)).

(3) Supra α-open soft set if (F,E)⊆̃ints(cls(ints(F,E))).

(4) Supra β-open soft set if (F,E)⊆̃cls(ints(cls(F,E))).

(5) Supra b-open soft set if (F,E)⊆̃cls(ints(F,E))∪̃ints(cls(F,E)).

(6) Supra A-soft set if (F,E) = (G,E) − (H,E) where (G,E) is supra open soft and
(H,E) is supra regular open soft set in X.

The set of all supra pre open (resp. semi open, α-open, β-open, b-open, A-) soft
sets is denoted by supra-POS(X) (resp. supra-SOS(X), supra-αOS(X), supra-βOS(X),
supra-BOS(X), supra-AS(X)) and the set of all supra pre closed (resp. semi closed, α-
closed, β-closed, b-closed ) soft sets is denoted by supra-PCS(X) (resp. supra-SCS(X),
supra-αCS(X), supra-βCS(X), supra-BCS(X)).

Definition 9 (13). Let (X,µ,E) be a supra soft topological space over X and (F,E) ∈
SS(X)E. Then, the supra P -soft interior of (F,E), denoted by intsP (F,E) is the soft
union of all supra pre-open soft subsets of (F,E) i.e

intsP (F,E) =
⋃̃{

(G,E) : (G,E) is a supra pre-open soft set and (G,E)⊆̃(F,E)
}
.



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 838

Also, the supra P -soft closure of F , denoted by clsP (F,E) is the soft intersection of all
supra pre-closed super soft sets of (F,E) i.e

clsP (F,E) =
⋂̃{

(H,E) : (H,E) is a supra pre-closed soft set and (F,E)⊆̃(H,E)
}
.

Definition 10. [1] A soft set (F,E) is called supra soft locally closed in a supra soft
topological space (X,µ,E) if (F,E) = (G,E)∩̃(H,E) where (G,E) is supra open soft and
(H,E) is supra closed soft in X. We will denote the family of all supra soft locally closed
sets of a supra soft topological space X by supra-SLC(X).

Definition 11 (1). A soft subset (F,E) of a supra soft topological space (X,µ,E) is called
supra soft dense set if cls(F,E) = X̃.

Definition 12 (1). A supra soft topological space (X,µ,E) is called supra soft submaximal
if every supra soft dense subset of (X,µ,E) is supra open soft.

Definition 13 (1,3,13). Let (X, τ1, A) and (Y, τ2, B) be soft topological spaces. Let µ1 be
an associated supra soft topology with τ1. Let u : X → Y and p : A→ B be mappings. Let
fpu : SS(X)A → SS(Y )B be a function. Then, fpu is called:

(1) Supra soft continuous if f−1pu (G,B) ∈ µ1 ∀ (G,B) ∈ τ2.

(2) Supra soft pre-continuous if f−1pu (G,B) ∈ supra-POS(X) ∀ (G,B) ∈ τ2.

(3) Supra soft semi-continuous if f−1pu (G,B) ∈ supra-SOS(X) ∀ (G,B) ∈ τ2.

(4) Supra soft α-continuous if f−1pu (G,B) ∈ supra-αOS(X) ∀ (G,B) ∈ τ2.

(5) Supra soft β-continuous if f−1pu (G,B) ∈ supra-βOS(X) ∀ (G,B) ∈ τ2.

(6) Supra soft B-continuous if f−1pu (G,B) ∈ supra-BOS(X) ∀ (G,B) ∈ τ2.

(7) Supra soft A-continuous function if f−1pu (G,B) ∈ supra−AS(X) ∀ (G,B) ∈ τ2.

(8) Supra soft locally closed continuous function (supra SLC-continuous) if f−1pu (G,B) ∈
supra− SLC(X) ∀ (G,B) ∈ τ2.

Definition 14 (2). Let (F,E) be a soft subset of a supra soft topological space (X,µ,E)
such that (F,E) = (G,E)∩̃(H,E), then (F,E) is said to be supra soft α-locally closed if
(G,E) is a supra α-open soft and (H,E) is a supra α-closed soft in X. We will denote
the family of all supra soft α-locally closed sets of a supra soft topological space X by
supra-SαLC(X).



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 839

3. Supra Soft Pre-Locally Closed Sets

In this section, we introduce the notion of supra soft P -locally closed sets in supra soft
topological spaces and discuss its relationships with other supra open soft sets in detail,
supported by counterexamples. Also, the notions of supra soft P ∗-locally closed sets and
supra soft P ∗∗-locally closed sets are introduced and studied.

Definition 15. Let (F,E) be a soft subset of a supra soft topological space (X,µ,E) such
that (F,E) = (G,E)∩̃(H,E), then (F,E) is said to be:

(1) Supra soft P -locally closed if (G,E) is a supra pre-open soft and (H,E) is a supra
pre-closed soft in X.

(2) Supra soft P ∗-locally closed if (G,E) is a supra pre-open soft and (H,E) is a supra
closed soft in X.

(3) Supra soft P ∗∗-locally closed if (G,E) is a supra open soft and (H,E) is a supra
pre-closed soft in X.

We will denote the family of all supra soft P -locally (resp. P ∗-locally and P ∗∗-locally)
closed sets of a supra soft topological space X by supra-SPLC(X) (resp. supra-SP ∗LC(X)
and supra-SP ∗∗LC(X))

Remark 1. A soft subset (F,E) of (X,µ,E) is supra soft P -locally (resp. P ∗-locally,
P ∗∗-locally) closed if its relative complement (F,E)c is the soft union of a pre-supra open
soft set and a supra pre-closed soft set (resp. a supra pre-closed soft set and a supra open
soft set, a supra closed soft set and a supra pre-open soft set).

In a supra soft topological space (X,µ,E), every supra soft P ∗- (resp. P ∗∗-) locally
closed is a supra soft P -locally closed.

Proof. Obvious from the fact that, every supra open (resp. closed) soft set is a supra
pre-open (resp. pre-closed) soft [[13], Theorem 5.1 (1)].

Remark 2. The converse of the above theorem is not true in general as shall shown in
the following example.

Example 1. [2] Suppose that there are four houses in the universe X given by X =
{a, b, c, d}. Let E = {e1, e2} be the set of decision parameters which stand for ”green
surroundings” and ”wooden” respectively.
Let (F1, E), (F2, E), (F3, E), (F4, E), (F5, E) be five soft sets over the common universe X
which describe the composition of the houses defined as follows:
F1(e1) = {a, c}, F1(e2) = {b, c},
F2(e1) = {b, c}, F2(e2) = {a, c},
F3(e1) = {a, b, c}, F3(e2) = {a, b, c},
F4(e1) = {a, b, d}, F4(e2) = {a, b, d},
F5(e1) = {b, c, d}, F5(e2) = {b, c, d}.



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 840

Hence, µ = {X̃, ϕ̃, (F1, E), (F2, E), (F3, E), (F4, E), (F5, E)} is a supra soft topology over
X. Therefore, the soft set (A,E) is supra soft P -locally closed in (X,µ,E), but not supra
soft P ∗-locally closed, where
A(e1) = {b}, A(e2) = {a}.
Also, the soft set (B,E) is supra soft P -locally closed in (X,µ,E), but not supra soft P ∗∗-
locally closed, where
B(e1) = {a, c, d}, A(e2) = {b, c, d}.

In a supra soft topological space (X,µ,E), every supra soft α-locally closed is a supra
soft P -locally closed.

Proof. Obvious from the fact that, every supra α-open (resp. α-closed) soft set is a
supra pre-open (resp. pre-closed) soft [[13], Theorem 5.2. (4)]

Remark 3. The converse of the above theorem is not true in general as shall shown in
the following example.

Example 2. In Example 1, the soft set (Z,E) is supra soft P -locally closed in (X,µ,E),
but not supra soft α-locally closed, where
Z(e1) = {a, b}, Z(e2) = {a, d}.

In a supra soft topological space (X,µ,E), every supra soft locally closed is a supra
soft P - (resp. P ∗- and P ∗∗-) locally closed.

Proof. Obvious from [[13], Theorem 5.1 (1)].

Remark 4. The converse Theorem 3 is not true in general as shall shown in the following
examples.

(1) In Example 1, the soft set (G,E) is a supra soft P -locally closed in (X,µ,E), but not
supra soft locally closed, where G(e1) = {a, c, d}, G(e2) = {b, c, d}.

(2) In Example 1, the soft set (H,E) is a supra soft P ∗-locally closed in (X,µ,E), but
not supra soft locally closed, where H(e1) = {a}, H(e2) = {d}.

(3) In Example 1, the soft set (K,E) is a supra soft P ∗∗-locally closed in (X,µ,E), but
not supra soft locally closed, where K(e1) = {a}, K(e2) = {b, c}.

Let (X,µ,E) be a supra soft topological space. Then, (F,E) is supra soft P -locally
closed if and only if (F,E) = (G,E)∩̃clsP (F,E) for some supra pre-open soft set (G,E).

Proof. Necessity:Let (F,E) be a supra soft P -locally closed set in X. Then, (F,E) =
(G,E)∩̃(H,E) where (G,E) is supra pre-open soft and (H,E) is supra pre-closed soft
in X. It follows, clsP (F,E)⊂̃clsP (H,E) = (H,E), where clsP (F,E) is a supra pre-closed
soft set. Therefore, (F,E)⊆̃(G,E)∩̃clsP (F,E)⊆̃(G,E)∩̃(H,E) = (F,E). Thus, (F,E) =
(G,E)∩̃clsP (F,E).
Sufficient: Follows directly from Definition 15 (1).

Let (F,E) be a subset of a supra soft topological space (X,µ,E). Then, the following
are equivalent:



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 841

(i) (F,E) ∈ supra-SPLC(X).

(ii) clsP (F,E)− (F,E) is supra pre-closed soft.

(iii) (F,E)∪̃[clsP (F,E)]c̃ is supra pre-open soft.

Proof.

(i) ⇒ (ii) : Let (F,E) ∈ supra-SPLC(X). By Theorem 3, (F,E) = (G,E)∩̃clsP (F,E) for
some supra pre-open soft set (G,E). It follows, clsP (F,E)−(F,E) = clsP (F,E)∩̃[(G,E)∩̃clsP
(F,E)]c̃ = clsP (F,E)∩̃(G,E)c̃ is supra pre-closed soft from [[13], Theorem 4.1 (2)],
where (G,E)c̃ is supra pre-closed soft set. Thus, clsP (F,E) − (F,E) is supra pre-
closed soft.

(ii) ⇒ (i) : Assume that (A,E) = [clsP (F,E) − (F,E)]c̃. From (ii), (A,E) is supra pre-
open soft in X. Hence, (A,E)∩̃clsP (F,E) = [clsP (F,E)−(F,E)]c̃∩̃clsP (F,E) = (F,E).
Therefore, (F,E) ∈ supra-SPLC(X) from Theorem 3.

(ii) ⇒ (iii) : Since clsP (F,E) − (F,E) is supra pre-closed soft in X from (ii). Then,
[clsP (F,E)− (F,E)]c̃ = (F,E)∪̃[clsP (F,E)]c̃ is supra pre-open soft in X.

(iii) ⇒ (ii) : Obvious.

Let (X,µ,E) be a supra soft topological space. Then, (F,E) is supra soft P ∗-locally
closed if and only if (F,E) = (H,E)∩̃cls(F,E) for some supra pre-open soft set (H,E).

Proof. Necessity:Let (F,E) be a supra soft P ∗-locally closed set in X. Then, (F,E) =
(H,E)∩̃(G,E) where (H,E) is supra pre-open soft and (G,E) is supra closed soft in X. It
follows, cls(F,E)⊆̃cls(G,E) = (G,E). Hence, (F,E)⊆̃(H,E)∩̃cls(F,E)⊆̃(H,E)∩̃(G,E) =
(F,E). Thus, (F,E) = (H,E)∩̃cls(F,E).
Sufficient: Obvious from Definition 15 (2).

Let (F,E) be a subset of a supra soft topological space (X,µ,E). Then, the following
are equivalent:

(i) (F,E) ∈ supra-SP ∗LC(X).

(ii) cls(F,E)− (F,E) is supra pre-closed soft.

(iii) (F,E)∪̃[cls(F,E)]c̃ is supra pre-open soft.

Proof. It is similar to the proof of Theorem 3.
Let (X,µ,E) be a supra soft topological space. Then, (F,E) is supra soft P ∗∗-locally

closed if and only if (F,E) = (G,E)∩̃clsP (F,E) for some supra open soft set (G,E).
Proof. It is similar to the proof of Theorem 3.
Let (F,E) be a subset of a supra soft topological space (X,µ,E). If (F,E) ∈ supra-

SP ∗∗LC(X), then cls(F,E) − (F,E) is supra pre-closed soft and (F,E)∪̃[cls(F,E)]c̃ is
supra pre-open soft.

Proof. It is similar to the proof of Theorem 3.



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 842

Remark 5. The converse Theorem 3 is not true in general as shall shown in the following
examples.

Example 3. In Example 1, for the soft set (G,E), where G(e1) = {a, c, d}, G(e2) =
{b, c, d}, we have cls(G,E)− (G,E) = (C,E), where C(e1) = {b}, C(e2) = {a}, is a supra
pre-closed soft and (G,E)∪̃[cls(G,E)]c̃ = (G,E) is a supra pre-open soft. But, (F,E) 6∈
supra-SP ∗∗LC(X).

Remark 6. The relative complement of a supra soft P - (resp. P ∗- and P ∗∗-) locally
closed set need not to be a supra soft P - (resp. P ∗- and P ∗∗-) locally closed. The following
examples support our claim.

(1) In Examples 3 (1), the soft set (G,E) is a supra soft P -locally closed in (X,µ,E), but
its relative complement (G,E)c is not supra soft P -locally closed, where Gc(e1) =
{b}, Gc(e2) = {a}.

(2) Suppose that there are four phones in the universe X given by X = {a, b, c, d}. Let
E = {e1, e2} be the set of decision parameters which stand for ”cheap” and ”model”
respectively.
Let (F1, E), (F2, E), (F3, E), (F4, E), (F5, E), (F6, E), (F7, E), (F8, E), (F9, E), (F10, E)
be ten soft sets over the common universe X which describe the composition of the
phones defined as follows:
F1(e1) = {a, b}, F1(e2) = {a},
F2(e1) = {b}, F2(e2) = {b},
F3(e1) = {a, b}, F3(e2) = {a, b},
F4(e1) = {a, b, c}, F4(e2) = {a, b, c},
F5(e1) = {a}, F5(e2) = {a},
F6(e1) = {a, b, c}, F6(e2) = {a, c},
F7(e1) = {a, b, d}, F7(e2) = {a, b, c},
F8(e1) = {a, b}, F8(e2) = {a, b, c},
F9(e1) = {a, b}, F9(e2) = {a, c},
F10(e1) = X, F10(e2) = {a, b, c}.
Hence, µ = {X̃, ϕ̃, (F1, E), (F2, E), (F3, E), (F4, E), (F5, E), (F6, E), (F7, E), (F8, E), (F9, E),
(F10, E)} is a supra soft topology over X. Therefore, the soft set (H,E) is a supra
soft P ∗-locally closed set in (X,µ,E), where H(e1) = {b}, H(e2) = {b, c}, but
its relative complement (H,E)c is not supra soft P ∗-locally closed, where Hc(e1) =
{a, c, d}, Hc(e2) = {a, d}.

(3) In Examples 3 (3), the soft set (K,E) is a supra soft P ∗∗-locally closed in (X,µ,E), but
its relative complement (K,E)c is not supra soft P -locally closed, where Kc(e1) =
{b, c, d}, Kc(e2) = {a, d}.

In a supra soft topological space (X,µ,E), every supra A-soft set is a supra soft P -
(resp. P ∗- and P ∗∗-) locally closed.

Proof. Follows from the fact that, every supra regular closed soft set in a supra soft
topological space (X,µ,E) is a supra closed soft set [[25], Remark 3.2].



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 843

Remark 7. The converse of the above theorem is not true in general as shall shown in
the following examples.

Example 4. In Example 3 (2), the soft set (G,E) is supra soft locally closed set. Since
(G,E) = (F10, E)∩̃(H,E), where (F10, E) is supra open soft and (H,E) is supra closed
soft in X defined by H(e1) = {c, d}, H(e2) = {b, d}. Hence, it is a supra soft P - (resp.
P ∗- and P ∗∗-) locally closed from Theorem 3, where G(e1) = {c, d}, G(e2) = {b}. On the
other hand, it is not supra A-soft.

In a supra soft topological space (X,µ,E), every supra open (resp. closed) soft set is
a supra soft P - (resp. P ∗- and P ∗∗-) locally closed in X.

Proof. Obvious.

Remark 8. The converse of the above theorem is not true in general as shall shown in
the following example.

Example 5. In Examples 3, the soft sets (G,E) (resp. (H,E) and (K,E)) are supra soft
P - (resp. P ∗- and P ∗∗-) locally closed in (X,µ,E), but all of them neither supra open soft
nor supra closed soft in (X,µ,E).

Definition 16. A soft subset (F,E) of a supra soft topological space (X,µ,E) is called
supra soft pre-dense set if clsP (F,E) = X̃.

Proposition 1. A supra soft pre-dense set (F,E) is supra pre-open soft in (X,µ,E) if
and only if it is supra soft P -locally closed.

Proof. Immediate from Theorem 3 and Definition 16.

Definition 17. A supra soft topological space (X,µ,E) is called supra soft pre-submaximal
if every supra softpre-dense subset of (X,µ,E) is supra pre-open soft.

Corollary 1. A supra soft topological space (X,µ,E) is supra soft pre-submaximal if and
only if every soft subset of (X,µ,E) is supra soft P -locally closed.

Proof. Immediate from Proposition 1.
Every supra soft submaximal space (X,µ,E) is supra soft pre-submaximal space.
Proof. Let (X,µ,E) be a supra soft submaximal space and (F,E) ∈ µ. Then,

cls(F,E) = X̃. It follows, cls(F,E) = X̃⊆̃clsP (F,E), where (F,E) ∈ µ ⊆ supra-POS(X).
Hence, (X,µ,E) is a supra soft pre-submaximal space.

Remark 9. The converse of Theorem 3 is not true in general as shall shown in the
following example.

Example 6. Suppose that there are four watches in the universe X given by X =
{a, b, c, d}. Let E = {e1, e2} be the set of decision parameters which stand for ”model”
and ”cheap” respectively.
Let (F1, E), (F2, E), (F3, E), (F4, E), (F5, E) be four soft sets over the common universe X
which describe the composition of the watches defined as follows:



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 844

F1(e1) = {a, b}, F1(e2) = {a},
F2(e1) = {b, d}, F2(e2) = {b, d},
F3(e1) = {a, c, d}, F3(e2) = X,
F4(e1) = {a, b, d}, F4(e2) = {a, b, d}.
Hence, µ = {X̃, ϕ̃, (F1, E), (F2, E), (F3, E), (F4, E)} is a supra soft topology over X, which
is a supra soft pre-submaximal space. On the other hand, the soft set G,E, where
G(e1) = {a, c, d}, G(e2) = {a, b, c} is supra soft dense set but not supra open soft in
X. Hence, (X,µ,E) is not supra soft submaximal space.

For a supra soft topological space (X,µ,E) we have the following implications from
Theorems 3, 3, 3, 3 and [[3], Corollary 4.1]. These implications are not reversible.

supra-SPLC(X) ← supra-SP ∗LC(X)
˚ ↑ ↑
supra-SP ∗∗LC(X)← supra-SLC(X) ←− supra-AS(X)
˚ ↑ ↗ ↖

↘ ↓
supra-OS(X) −→ supra-αOS(X) −→ supra-SOS(X)
˚ ↓ ↙
supra-POS(X) −→ supra-BOS(X)

4. Decompositions of Supra Soft Continuity via Supra Pre-Open soft
sets

In this section, we introduce three different notions of generalized supra soft conti-
nuity, namely supra SPLC-continuous functions, supra SP ∗LC-continuous functions and
supra SP ∗∗LC-continuous functions. Furthermore, we obtain decompositions of supra soft
continuity. Finally, Several examples are provided to illustrate the behavior of these new
classes of soft functions.

Definition 18. Let (X, τ1, A) and (Y, τ2, B) be soft topological spaces. Let µ1 be an as-
sociated supra soft topology with τ1. Let u : X → Y and p : A → B be mappings. Let
fpu : SS(X)A → SS(Y )B be a function. Then, fpu is called:

(1) Supra soft pre-locally closed continuous function (supra SPLC-continuous) if f−1pu (G,B) ∈
supra-SPLC(X) ∀ (G,B) ∈ τ2.

(2) Supra soft pre∗-locally closed continuous function (supra SP ∗LC-continuous) if f−1pu (G,B) ∈
supra-SP ∗LC(X) ∀ (G,B) ∈ τ2.

(3) Supra soft P ∗∗-locally closed continuous function (supra SP ∗∗LC-continuous) if f−1pu (G,B) ∈
supra-SP ∗∗LC(X) ∀ (G,B) ∈ τ2.

Let (X, τ1, A) and (Y, τ2, B) be soft topological spaces. Let µ1 be an associated supra
soft topology with τ1. Let u : X → Y and p : A→ B be mappings. Let fpu : SS(X)A →



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 845

SS(Y )B be a function. Then, every supra SP ∗LC- (resp. supra SP ∗∗LC)-continuous
function is a supra SPLC-continuous.

Proof. It is obvious from Theorem 3.

Remark 10. The converse of Theorem 4 is not true in general, as shown in the following
examples.

(1) Let X = {a, b, c, d}, Y = {x, y, z}, A = {e1, e2} and B = {k1, k2}. Define u : X → Y
and p : A→ B as follows:
u(a) = {z}, u(b) = {y} , u(c) = {x}, u(d) = {x} and
p(e1) = {k2}, p(e2) = {k1}. Let (X, τ1, A) be a soft topological space over X where,
τ1 = {X̃, ϕ̃, (F1, A)}, where (F1, A) is a soft set over X defined as follows:
F1(e1) = {b, c}, F1(e2) = {a, c}.
Consider the supra soft topology µ1 = {X̃, ϕ̃, (F1, A), ......., (F5, A)} in Example 1.
Let (Y, τ2, B) be a soft topological space over Y where,
τ2 = {Ỹ , ϕ̃, (G,B)}, where (G,B) is a soft set over Y defined by:
G(k1) = {y}, G(k2) = {z}.
Let fpu : (X, τ1, A)→ (Y, τ2, B) be a soft function. Then, f−1pu ((G,B)) = {(e1, {b}), (e2, {a})}
is a supra soft P -locally closed in X, but not supra soft P ∗-locally closed. Hence,
fpu is a supra SPLC-continuous, but not supra SP ∗LC-continuous.

(2) In (1), let (Y, τ2, B) be a soft topological space over Y where,
τ2 = {Ỹ , ϕ̃, (G,B)}, where (G,B) is a soft set over Y defined by:
G(k1) = {x, z}, G(k2) = {x, y}.
Let fpu : (X, τ1, A)→ (Y, τ2, B) be a soft function. Then, f−1pu ((G,B)) = {(e1, {a, c, d}),
(e2, {b, c, d})} is a supra soft P -locally closed in X, but not supra soft P ∗∗-locally
closed. Hence, fpu is a supra SPLC-continuous, but not supra SP ∗∗LC-continuous.

Let (X, τ1, A) and (Y, τ2, B) be soft topological spaces. Let µ1 be an associated supra
soft topology with τ1. Let u : X → Y and p : A→ B be mappings. Let fpu : SS(X)A →
SS(Y )B be a function. Then, every supra SαLC-continuous function is a supra SPLC-
continuous.

Proof. It is obvious from Theorem 3.

Example 7. Let X = {a, b, c, d}, Y = {x, y, z, w}, A = {e1, e2} and B = {k1, k2}. Define
u : X → Y and p : A→ B as follows:
u(a) = {z}, u(b) = {w} , u(c) = {x}, u(d) = {y} and
p(e1) = {k2}, p(e2) = {k1}. Let (X, τ1, A) be a soft topological space over X where,
τ1 = {X̃, ϕ̃, (F1, A)}, where (F1, A) is a soft set over X defined as follows:
F (e1) = {a, b}, F (e2) = {a, b}.
Consider the supra soft topology µ1 in Example 1, µ1 = {X̃, ϕ̃, (F1, A), .......(F10, A)}. Let
(Y, τ2, B) be a soft topological space over Y where,
τ2 = {Ỹ , ϕ̃, (G,B)}, where (G,B) is a soft set over Y defined by:
G(k1) = {z, w}, G(k2) = {y, z}.



F. A. Gharib et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 835-849 846

Let fpu : (X, τ1, A)→ (Y, τ2, B) be a soft function. Then, f−1pu ((G,B)) = {(e1, {a, b}), (e2, {a, d})}
is a supra soft P -locally closed in X, but it is not supra soft α-locally closed. Hence, fpu
is a supra SPLC-continuous, but it is not supra soft SαLC-continuous.

Let (X, τ1, A) and (Y, τ2, B) be soft topological spaces. Let µ1 be an associated supra
soft topology with τ1. Let u : X → Y and p : A→ B be mappings. Let fpu : SS(X)A →
SS(Y )B be a function. Then, every supra SLC-continuous function is a supra SPLC-
(resp. supra SP ∗LC- and supra SP ∗∗LC)-continuous.

Proof. Follows from Theorem 3.

Remark 11. The converse of Theorem 4 is not true in general, as shown in the following
examples.

(1) In Examples 4 (2), fpu is a supra SPLC-continuous, but not supra SLC-continuous.

(2) Let X = {a, b, c, d}, Y = {x, y, z}, A = {e1, e2} and B = {k1, k2}. Define u : X → Y
and p : A→ B as follows:
u(a) = {z}, u(b) = {y} , u(c) = {y}, u(d) = {x} and
p(e1) = {k2}, p(e2) = {k1}. Let (X, τ1, A) be a soft topological space over X where,
τ1 = {X̃, ϕ̃, (F1, A)}, where (F1, A) is a soft set over X defined as follows:
F1(e1) = {b, c}, F1(e2) = {a, c}.
Consider the supra soft topology µ1 = {X̃, ϕ̃, (F1, A), ......., (F5, A)} in Example 1.
Let (Y, τ2, B) be a soft topological space over Y where,
τ2 = {Ỹ , ϕ̃, (G,B)}, where (G,B) is a soft set over Y defined by:
G(k1) = {z}, G(k2) = {x}.
Let fpu : (X, τ1, A)→ (Y, τ2, B) be a soft function. Then, f−1pu ((G,B)) = {(e1, {a}), (e2, {d})}
is a supra soft P ∗-locally closed in X, but not supra soft locally closed. Hence, fpu
is a supra SP ∗LC-continuous, but not supra SLC-continuous.

(3) In (2), let (Y, τ2, B) be a soft topological space over Y where,
τ2 = {Ỹ , ϕ̃, (G,B)}, where (G,B) is a soft set over Y defined by:
G(k1) = {z}, G(k2) = {y}.
Let fpu : (X, τ1, A)→ (Y, τ2, B) be a soft function. Then, f−1pu ((G,B)) = {(e1, {a}),
(e2, {b, c})} is a supra soft P ∗∗-locally closed in X, but not supra soft locally closed.
Hence, fpu is a supra SP ∗∗LC-continuous, but not supra SLC-continuous.

Let (X, τ1, A) and (Y, τ2, B) be soft topological spaces. Let µ1 be an associated supra
soft topology with τ1. Let u : X → Y and p : A→ B be mappings. Let fpu : SS(X)A →
SS(Y )B be a function. Then, every supra soft A-continuous function is a supra SPLC-
(resp. supra SP ∗LC- and supra SP ∗∗LC)-continuous.

Proof. It is obvious from Theorem 3.

Remark 12. The converse of Theorem 4 is not true in general, as shown in the following
example.



REFERENCES 847

Example 8. In Example 7, consider the supra soft topology µ1 in Example 3 (2), µ1 =
{X̃, ϕ̃, (F1, A), .......(F10, A)}. Let (Y, τ2, B) be a soft topological space over Y where,
τ2 = {Ỹ , ϕ̃, (G,B)}, where (G,B) is a soft set over Y defined by:
G(k1) = {x, y}, G(k2) = {w}.
Let fpu : (X, τ1, A)→ (Y, τ2, B) be a soft function. Then, f−1pu ((G,B)) = {(e1, {c, d}), (e2, {b})}
is a supra soft P - (resp. P ∗- and P ∗∗-) locally closed in X, but it is not supra A-soft.
Hence, fpu is a supra SPLC- (resp. supra SP ∗LC- and supra SP ∗∗LC)-continuous, but
it is not supra soft A-continuous.

For a supra soft topological space (X,µ,E) we have the following implications from
Theorems 4, 4, 4 and [[3], Corollary 6.1]. These implications are not reversible.

supra SPLC-continuity ← supra SP ∗LC-continuity
˚ ↑ ↑
supra SP ∗∗LC-continuity ← supra SLC-continuity ←− supra soft A-continuity
˚ ↑ ↗ ↖

↘ ↘
supra soft continuity −→ supra soft α-continuity −→ supra soft semi-continuity
˚ ↓ ↙
supra soft pre-continuity −→ supra soft B-continuity

5. Conclusion

The aim of this paper, is to introduce new types of soft sets in supra soft topological
spaces called, supra soft P -locally closed sets, supra soft P ∗-locally closed sets and supra
soft P ∗∗-locally closed sets. Also, new types of soft continuity are introduced. Further-
more, some of their basic properties are obtained. In future, the generalization of these
concepts by using soft ideals notion [15] will be introduced and the future research will be
undertaken in this direction.

Acknowledgments

The authors gratefully acknowledge the approval and the support of this research from
the Deanship of Scientific Research study by the grant no. 8-068-435, K. S. A., Northern
Border University, Arar.

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