Analyzing the Locus of Soft Spheres: Illustrative Cases and Drawings


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 4, 2018, 946-957
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Analyzing the Locus of Soft Spheres: Illustrative Cases
and Drawings

Güzide Şenel

Department of Mathematics, Faculty of Arts and Science, Amasya University, Amasya,
Turkey

Abstract. In the studies that have been carried out so far, the definition of soft sphere has been
made, although it has been accepted very hard to revive it mathematically. In this study, the
differences among soft real numbers and soft points are used for the first time to revive the soft
sphere mathematically thus the locus of the soft sphere can be analyzed. Applications of soft
spheres are supported by suitable examples to discuss the locus of them with scrupulous attention
to detail.

2010 Mathematics Subject Classifications: 03F03, 03B10, 03E75, 54E35

Key Words and Phrases: Soft set, Soft real number, Soft point, Soft sphere, Soft topology, Soft
metric

1. Introduction

In some theories such as fuzzy set, rough set, vague set, intuitionistic fuzzy set and
interval mathematics complicated problems can not be solved because of not being free
from the parametrization tool of the theory.
Molodtsov [10] provided a general framework with the involvement of parameters and
paved the way for new valuable works such as [1–4, 10–15].
The topological structures of set theories dealing with uncertainities were first defined by
Chang [5]. In this study, fuzzy topology is introduced and its properties are studied. The
notion of soft topological spaces introduced by Shabir and Naz [9] on an initial universe
with a fixed set of parameters.
For a deeper discussion of soft topological spaces, more specialized notions from soft
topological space theory were introduced in [6] and [7] as needed. We shall discuss this
again at somewhat greater length in the 2. Section. These development studies on soft
topological spaces motivated researchers to introduce the notion of soft metric space.

In the early 21th century, attempts to develop appropriate conceptual frameworks for
dealing with soft topological structure led Das and Samanta [7] to the definition of soft

DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3321

Email address: g.senel@amasya.edu.tr (G. Şenel)

http://www.ejpam.com 946 c© 2018 EJPAM All rights reserved.



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 947

metric spaces. Metric space is widely used in mathematics and soft metric space is a
generalization of metric spaces. That is to say, soft metric spaces can be developed in a
metric view with soft sets. While these important results caused most mathematicians
to think of soft metric spaces as just a rather convenient tool to define and to deal with
soft topological spaces, a few began to study soft metric spaces for their own sake as
[11]. The development of the theory of soft metric spaces has proceeded in the following
main directions: General theory of soft metric spaces and soft topological theory of metric
spaces. For each of the cases the simplest and most fruitful method which the soft metric
proposed was the introduction of the notion of soft distance and soft spheres. Rather than
discuss the soft metric spaces in full generality, let us look at a particular situation of the
view of soft spheres.

One of the most important component of soft metric spaces is soft spheres. Spherical
structure plays an important role in the study of soft metric space’s properties. Generally
speaking, the soft spherical structure in soft metric space is equivalent to the soft open
sets in a soft topological space.

Up to today, studies of soft metric spaces are made without trying to analyze the
meaning of soft spheres. The searchers on soft spheres just mere ”presentation” of the
underlying soft metric spaces. I suspect that more can be said in specifically about which
soft spheres can be obtained from soft metric space, but the researchers explore this matter
further here.

The main source of this paper inspiration is to achieve the structure of soft sphere
in detail. The motivate is to gain in interest if the locus of the sphere is revived math-
ematically. Thus it is reasonable to attempt, using the soft real number and soft point,
to achieve the structure of soft spheres. In the last section this main result is stated and
proved by soft sphere drawings that is shed a light on analyzing the locus of soft spheres.

In one of our applications we describe a more interesting example featuring a soft metric
that is not given a soft cyclic sphere. Hence we show different shapes of soft spheres. It is
obvious that this result shows a soft sphere supplies an algorithm to effectively recognize
soft points in R̃∗(E).

The aim of this article is to study the relationship between the size of soft spheres,
as measured by its defined soft metric, and the extent to which the soft sphere fails to
be defined in different soft metric. It seems that the relations between soft open and soft
closed spheres emerge most clearly when the setting is quite abstract, and this motivates
our approach to the subject.

2. Background in Soft Sets and Soft Metric Spaces

In the course of writing this paper I learned that the authors have simultaneously
obtained results similar to each other in certain respects as the proofs are similar in spirit
to that of each other. In the end of the literature search, nonetheless, the references that
are written below were my main source of inspiration:

I recall some basic notions of soft set theory which may be found in [1–3, 10, 11] for
further details. For a deeper discussion of soft point, I refer the reader to [12]. Also,



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 948

for detailed definitions, propositions and theorems about soft positive and negative real
numbers, the reference [6] can be examined. This case has been thoroughly studied; see
[7] for more details, theorems and examples.

For a comprehensive treatment and for references to the extensive literature on the
related subject in this article one may refer to these references to related subject.

Throughout this work, U refers to an initial universe, E is a set of parameters, A⊆E
and P (U) is the power set of U .

Definition 1. ([3, 10]) A soft set fA on the universe U is a set defined by

fA : E → P(U) such that fA(x) = ∅ if x /∈ A.

Here fA is also called an approximate function. A soft set over U can be represented by
the set of ordered pairs

fA =
{(
x, fA(x)

)
: x ∈ E, fA(x) ∈ P (U)

}
.

Note that the set of all soft sets over U will be denoted by S.

Definition 2. ([3]) Let fA ∈ S. Then,
If fA(x) = ∅ for all x ∈ E, then fA is called an empty soft set, denoted by Φ.
If fA(x) = U for all x ∈ E, then fA is called universal soft set, denoted by Ẽ.
If fA(x) is a singleton set for all x ∈ E, then fA is called a singleton soft set.

Definition 3. ([6]) Let X be a non-empty set and E be a non-empty parameter set. Then
a function ε : E → X is said to be a soft element of X. A soft element ε of X is said to
belong to a soft set A of X, denoted by ε ∈ A, if ε(e) ∈ A(e), ∀e ∈ E. Thus a soft set A
of X with respect to the index set E can be expressed as A(e) = {ε(e), ε ∈ A}, e ∈ E.

This chapter continues to constitute sufficient preparation by giving definitions of soft
real number and soft point:

Definition 4. ([6]) Let R be the set of real numbers, P(R) be the collection of all non-
empty bounded subsets of R, E be a set of parameters and A ⊆ E. Then a mapping
f : A → P(R) is called a soft real set. It is denoted by (f,A) . If in particular (f,A) is
a singleton soft set, then identifying (f,A) with the corresponding soft element, it will be
called a soft real number.

Now on, the single parameter set to be used is denoted by E to define soft sets and
their operations. Thus, the subscript E can be deleted from the soft sets (f,E), i.e, a
soft set fE will be denoted shortly by f unless this will cause confusion. In general, soft
sets and their approximate functions are denoted by f, g, h, ..., and soft sets and their
approximate functions are used interchangeably.

Definition 5. ([6]) Let us denote the collection of all soft points of f by SP (f), the set
of all soft real sets by R(E), the set of all soft real numbers by R̃(E) and the set of all



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 949

non-negative soft real numbers by R̃∗(E). If a soft real set is a singleton soft set, it will
be called a soft real number and denoted by r̃, s̃ etc.

The best general reference about aritmetic operations of soft real sets and soft real
numbers is [6] for a fuller treatment.

Definition 6. ([2]) The soft set f is called a soft point in S, if for the parameter ei ∈ E
such that f(ei) 6= ∅ and f(ej) = ∅, for all ej ∈ E \ {ei} is denoted by

(
eif
)
j

for all

i, j ∈ N+.
Note that the set of all soft points of f will be denoted by SP (f).

Definition 7. ([12]) Let f ∈ S. Then,
If f(e) = ∅ for all e ∈ E, then f is called null soft point, denoted by eΦ.
If f(e) = U for all e ∈ E, then f is called universal soft point, denoted by eẼ.
If there is only one parameter e ∈ E in f , then f is denoted by efi. If there is only

one parameter e ∈ E in f , and f(e) = {u} then f is denoted by ef .

Definition 8. ([12]) The soft point
(
eif
)
j

is said to belong to a soft set g ∈ S, denoted by(
eif
)
j
∈̃ g, if for the parameter ei ∈ E and f(ei) ⊆ g(ei).

Proposition 1. ([12])

(i) f⊆̃g, if
(
eif
)
j
∈̃f then

(
eif
)
j
∈̃ g for all ei∈̃f .

(ii) f = g, if and only if
(
eif
)
j
∈̃f then

(
eif
)
j
∈̃ g and

(
eif
)
j
∈̃ g and

(
eif
)
j
∈̃f for all

ei∈̃f .

Definition 9. ([11]) Let ∅ 6= X ⊆ E, f ∈ SX(U) and f : X → P(U) be one to one
function. We denote by R̃∗(E) the set of all soft real numbers and fi, fj , fs ∈ SX(U) ;(
eif
)
j
,
(
exf
)
y
,
(
esf
)
k
∈̃f . A mapping

d :E × E → R̃∗(E)((
eif
)
j
,
(
exf
)
y

)
→ d̃

((
eif
)
j
,
(
exf
)
y

)
is said to be a soft metric on the soft set f if d̃ satisties the following conditions:

(i) d̃
((
eif
)
j
,
(
exf
)
y

)
≥ 0.

(ii) d̃
((
eif
)
j
,
(
exf
)
y

)
= 0⇔

(
eif
)
j

=
(
exf
)
y

(iii) d̃
((
eif
)
j
,
(
exf
)
y

)
= d̃
((
exf
)
y
,
(
eif
)
j
)

(iv) d̃
((
eif
)
j
,
(
exf
)
y

)
≤ d̃
((
eif
)
j
,
(
esf
)
k
) + d̃

((
esf
)
k
,
(
exf
)
y
)



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 950

If d̃ is a soft metric on the soft set f then, f is called soft metric space and denoted by
(f, d̃).

Definition 10. ([11]) Let
(
eif
)
j

and
(
exf
)
y

be soft points of a soft metric space. The

value of d̃
((
eif
)
j
,
(
exf
)
y

)
is called as the soft distance between the soft points

(
eif
)
j

and(
exf
)
y

.

Definition 11. ([7]) Let (f, d̃) be a soft metric space. g 6= Φ and g⊆̃f . Then the diameter
of g is denoted by δ(g) and is defined by

d
(
(g)(γ)

)
= sup

{
d̃
((
eif
)
j
,
(
exf
)
y

)
;
(
eif
)
j
,
(
exf
)
y
∈̃ g
}

, ∀γ ∈̃f .

Definition 12. ([7]) Let (f, d̃) be a soft metric space,
(
exf
)
y

be a fixed soft point of f

and g ⊆ f . Then the distance of the soft point
(
exf
)
y

from the soft set g is denoted by

δ(
(
exf
)
y
, g) and defined by

δ(
(
exf
)
y
, g)(γ) = inf

{
d̃
((
exf
)
y
,
(
eif
)
j

)
(γ);

(
eif
)
j

)
∈̃ g
}

Definition 13. ([7]) Let (f, d̃) a soft metric space, r̃ be a non-negative soft real number
and

(
eif
)
j
,
(
exf
)
y
∈̃ f .

B̃d̃
((
exf
)
y
, r̃
)

=
{(
eif
)
j
∈̃ f : d̃

((
eif
)
j
,
(
exf
)
y

)
<̃ r̃
}

B̃d̃
[(
exf
)
y
, r̃
]

=
{(
eif
)
j
∈̃ f : d̃

((
eif
)
j
,
(
exf
)
y

)
≤̃ r̃
}

K̃d̃

((
exf
)
y
, r̃
)

=
{(
eif
)
j
∈̃ f : d̃

((
eif
)
j
,
(
exf
)
y

)
=̃ r̃
}

These soft sets are called respectively soft open sphere, soft closed sphere and soft k-
sphere with center

(
exf
)
y

and radius r̃.

For the sake of clarity, the notation can be written as B̃d̃
((
exf
)
y
, r̃
)

= B̃
((
exf
)
y
, r̃
)
.

Besides, there is a property of soft spheres that is valid for all soft metrics:

For r̃1 < r̃2,

B̃
((
exf
)
y
, r̃1

)
⊂̃ B̃

[(
exf
)
y
, r̃1

]
⊂̃ B̃

((
exf
)
y
, r̃2

)
⊂̃ B̃

[(
exf
)
y
, r̃2

]
It is obvious from the definition of soft metric.



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 951

3. More on Soft Spheres with Using Soft Real Numbers and Soft Points

In this section, the most important facts about soft spheres are compiled for the purpose
of exposition the structure of soft spheres are studied in [13].

Proposition 2. ([13]) Any soft intersection of soft open spheres need not to be soft open.
The following example shows that:

Example 1. ([13]) Let me show the equation
∞⋂
n̄=1̄

B̃
((
exf
)
y
, 1̄ + 1̄

n̄

)
= B̃

((
exf
)
y
, 1̄
)

.

a)
∞⋂
n̄=1̄

B̃
((
exf
)
y
, 1̄ + 1̄

n̄

)
⊂̃ B̃

((
exf
)
y
, 1̄
)

(?)(
eif
)
j
∈̃
∞⋂
n̄=1̄

B̃
((
exf
)
y
, 1̄ + 1̄

n̄

)
⇒
(
eif
)
j
∈̃B

((
exf
)
y
, 1̄ + 1̄

n̄

)
⇒ |̄i− x̄| <̃ 1̄ + 1̄

n̄

(∀ε > 0 ), ī <̃ j̄ + ε , ī ≥̃ j̄
(∀ε > 0) ( 1̄

n̄ <̃ ε) : ∃n̄ ∈̃N

}
⇒ |̄i− x̄| <̃ 1̄ + 1̄

n̄≤̃1̄⇒
(
eif
)
j
∈̃ B̃

((
exf
)
y
, 1̄
)

.

b)(
eif
)
j
∈̃B̃
((
exf
)
y
, 1̄
)
⇒ρ̃
((
eif
)
j
,
(
exf
)
y

)
≤̃1̄<̃1̄ +

1̄

n̄
, (∃n̄∈̃N+)

ρ̃
((
eif
)
j
,
(
exf
)
y

)
<̃1̄ +

1̄

n̄
⇒
(
eif
)
j
∈̃
∞⋂
n̄=1̄

B̃
((
exf
)
y
, 1̄ +

1̄

n̄

)
Proposition 3. ([13]) Any soft union of soft closed spheres need not to be soft closed.
The following example shows that:

Example 2. ([13]) Let me show the equation
∞⋃
n=1

B̃
((
exf
)
y
, n̄

¯n+1

)
=̃B

((
exf
)
y
, 1̄
)

.

a)
∞⋃
n=1

B̃
((
exf
)
y
, n̄

¯n+1

)
⊂̃B

((
exf
)
y
, 1̄
)

(?)

∃
(
exf
)
y
∈̃B̃
(
exf
)
y
, n̄

¯n+1

)
⇒ ρ̃

((
exf
)
y
, (eif

)
j

)
≤̃ n̄

¯n+1
<̃1̄ (∀n̄∈̃N+)

⇒ ρ̃
((
exf
)
y
, (eif

)
j

)
<̃1̄⇒

(
eif
)
j
∈̃Bρ

((
exf
)
y
, 1̄
)

.

b)
(
eif
)
j
∈̃ B̃

(
exf
)
y
, 1̄
)
⇒
(
eif
)
j
∈̃
∞⋃
n̄=1̄

B̃
((
exf
)
y
, n̄

¯n+1

)
(?)(

eif
)
j
∈̃B̃
(
exf
)
y
, 1̄
)
⇒ ρ̃

((
exf
)
y
, (eif

)
j

)
<̃1̄

Suppose that, (exf
)
y
< (eif

)
j
< (exf

)
y

+ 1̄
n̄ and (1/n̄ = ε).

Then, (exf
)
y
< (eif

)
j
≤ (exf

)
y
. So, (exf

)
y

= (eif
)
j
.

This contradicts our assumption and leads to the contradiction that
(
eif
)
j
∈̃ B̃

(
exf
)
y
, 1̄
)

.

Hence,



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 952

(exf
)
y

+ 1̄
n̄ < (eif

)
j

⇒ (eif
)
j
< n̄

¯an+1
≤ n̄

¯n+1

⇒ (exf
)
y
< (eif

)
j
≤ n̄

¯n+1

⇒ (eif
)
j
∈̃
(

(exf
)
y
, n̄

¯n+1

)
⇒ ρ̃

((
exf
)
y
, (eif

)
j

)̃
≤ n̄

¯n+1
.

The benefit of formulating the notion of ‘spheres’ as above will become clear with
applications in the next section:

4. Analizing The Locus of the Soft Spheres with Examples and
Drawings

This section is one of the major steps in analizing the locus of the soft spheres has been
accepted very hard to revive until now. For this purpose, different soft metrics are defined
and obtained a soft metric that is not given a soft cyclic sphere. Results are proved that
a soft sphere supplies an algorithm to effectively recognize soft points in R̃∗(E).

Example 3. Let d1 function is defined as d̃1

((
eif
)
j
,
(
exf
)
y

)
=
√

(̄i− x̄)2 + (j̄ − ȳ)2 in f .
Then, show the following cases:

(i) (f, d̃1) is a soft metric space.

(ii) Find the soft sets B̃d̃1 ((0̄, 0̄), 2̄), B̃d̃1 [(1̄, 1̄), 3̄] and K̃d̃1
((1̄, 2̄), 1̄).

(iii) Draw the soft spheres that are found in (2).

Solution:

d̃1 :E × E → R̃∗(E)((
eif
)
j
,
(
exf
)
y

)
→ d̃1

((
eif
)
j
,
(
exf
)
y

)
=
√

(̄i− x̄)2 + (j̄ − ȳ)2

(m1) The square root function is positive defined. Hence, this function is positive, too.

(m2) d̃1

((
eif
)
j
,
(
exf
)
y

)
= 0̄⇔

(
eif
)
j

=
(
exf
)
y

(?)√
(̄i− x̄)2 + (j̄ − ȳ)2 = 0̄⇒ (̄i− x̄)2 + (j̄ − ȳ)2 = 0̄. Then, (̄i− x̄)2 = 0̄ and (j̄ − ȳ)2 = 0̄.

(m3) d̃
((
eif
)
j
,
(
exf
)
y

)
= d̃
((
exf
)
y
,
(
eif
)
j
) (?)

If square function is applied both of sides, it is obvious.

(m4) ∀
(
eif
)
j
,
(
exf
)
y
,
(
esf
)
k
∈̃f



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 953

Figure 1

d̃
((
eif
)
j
,
(
exf
)
y

)
≤ d̃
((
eif
)
j
,
(
esf
)
k

)
+ d̃
((
esf
)
k

)
,
(
exf
)
y

))
B̃d̃1 ((0̄, 0̄), 2̄) =

{(
eif
)
j
∈̃ f |d̃1((0̄, 0̄), (̄i, j̄)) <̃ 2̄

}
=
{(
eif
)
j
∈̃ f |

√
(0̄− ī)2 + (0̄− j̄)2 <̃ 2̄

}
=
{(
eif
)
j
∈̃ f |̄i2 + j̄2 <̃ 4̄

}
, r̃ = 2̄ and O(0̄, 0̄).

Figure 2

Kd̃1
((1̄, 2̄), 1̄) =

{(
eif
)
j
∈̃ f |d̃1((1̄, 2̄), (̄i, j̄)) = 1̄

}
=
{(
eif
)
j
∈̃ f |

√
(̄i− 1̄)2 + (j̄ − 2)2 = 1̄

}
=
{(
eif
)
j
∈̃ f |(̄i− 1̄)2 + (j̄ − 2̄)2 = 1̄

}
r̃ = 1̄ and O(1̄, 2̄).

Figure 3



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 954

We now describe a more interesting example featuring a metric that is not posed a
cyclic sphere:

Example 4. Let d2 function is defined as d̃2

((
eif
)
j
,
(
exf
)
y

)
= |̄i− x̄|+ |j̄− ȳ| in f . Then,

show the following cases:

(i) (f, d̃2) is a soft metric space.

(ii) Find the soft sets B̃d̃2 ((1̄, 1̄), 2̄), B̃d̃2 [(0̄, 0̄), 1̄] and K̃d̃2
((0̄, −̄2), 1̄).

(iii) Draw the soft spheres that are found in (2).

Solution:

d̃2 :E × E → R̃∗(E)

((
eif
)
j
,
(
exf
)
y

)
→ d̃2

((
eif
)
j
,
(
exf
)
y

)
= |̄i− x̄|+ |j̄ − ȳ| ≥̃ 0̄

(m2) d̃2

((
eif
)
j
,
(
exf
)
y

)
= 0̄⇔ |̄i− x̄|+ |j̄ − ȳ| = 0̄⇔ |̄i− x̄| = 0̄ and |j̄ − ȳ| = 0̄

(m4) ∀
(
eif
)
j
,
(
exf
)
y
,
(
esf
)
k
∈̃ f

d̃2

((
eif
)
j
,
(
exf
)
y

)
= |̄i− x̄|+ |j̄ − ȳ|

= |̄i− s̄+ s̄− x̄|+ |j̄ − k̄ + k̄ − ȳ|
≤̃|̄i− s̄|+ |s̄− x̄|+ |j̄ − k̄|+ |k̄ − ȳ|
= d̃2

((
eif
)
j
,
(
esf
)
k

)
+ d̃2

((
esf
)
k
,
(
exf
)
y

)

B̃d̃2 ((1̄, 1̄), 2̄) =
{(
eif
)
j
∈̃ f |d̃2((1̄, 1̄), (̄i, j̄)) <̃ 2̄

}
=
{(
eif
)
j
∈̃ f ||1̄− ī|+ |1̄− j̄| <̃ 2̄

}
Critical points : |1̄− ī|+ |1̄− j̄| <̃ 2̄⇒ ī = 1̄ and j̄ = 1̄
(i) ī <̃ 1̄ and j̄ <̃ 1̄⇒ 1̄− ī+ 1̄− j̄ <̃ 2̄⇒ ī+ j̄ > 0̄ or
(ii) ī <̃ 1̄ and j̄ <̃ 1̄⇒ −1̄ + ī− 1̄ + j̄ <̃ 2̄⇒ ī+ j̄ <̃ 4̄ or
(iii) ī <̃ 1̄ and j̄ <̃ 1̄⇒ −1̄ + ī+ 1̄− j̄ <̃ 2̄⇒ ī− j̄ <̃ 2̄ or



Güzide Şenel / Eur. J. Pure Appl. Math, 11 (4) (2018), 946-957 955

(iv) ī <̃ 1̄ and j̄ <̃ 1̄⇒ 1̄− ī− 1̄ + j̄ <̃ 2̄⇒ j̄ − ī <̃ 2̄ or
(v) ī = 1̄⇒ |1̄− j̄| <̃ 2̄⇒ −2̄ <̃ 1̄− j̄ <̃ 2̄⇒ −1̄ <̃ j̄ <̃ 3̄ or
(vi) j̄ = 1̄⇒ |1̄− ī| <̃ 2̄⇒ −1̄ <̃ ī <̃ 3̄.

We shall draw heavily on ideas from these soft real numbers:

Figure 4

K̃d̃2
((0̄, −̄2), 1̄) =

{(
eif
)
j
∈̃ f |d̃2((0̄, −̄2), (̄i, j̄)) = 1̄

}
=
{(
eif
)
j
∈̃ f ||̄i|+ |j̄ + 2̄| = 1̄

}

It is obvious that the above k-sphere supplies an algorithm to effectively recognize soft
points in f that can be seen on the drawing:
(i) ī <̃ 0 and j̄ <̃ − 2̄⇒ ī+ j̄ + 2̄ = 1̄⇒ ī+ j̄ = −1̄ or
(ii) ī <̃ 0̄ and j̄ <̃ − 2̄⇒ −ī− j̄ − 2̄ = 1̄⇒ ī+ j̄ = −3̄ or
(iii) ī <̃ 0̄ and j̄ <̃ − 2̄⇒ ī− j̄ − 2̄ = 1̄⇒ ī− j̄ = 3̄ or
(iv) ī <̃ 0̄ and j̄ <̃ − 2̄⇒ −ī+ j̄ + 2̄ = 1̄⇒ j̄ − ī = −1̄ or
(v) ī = 0̄⇒ |̄i+ 2̄| = 1̄⇒ j̄ = −1̄, j̄ = −3̄ or
(vi) j̄ = −2̄⇒ |̄i| = 1̄⇒ ī = 1̄, ī = −1̄.

Figure 5

The control of the radius is made and it is equal to 1:

d̃2

(
(0̄,−2̄),

(
−1̄
2̄
, −5̄

2̄

))
=
∣∣∣−1̄

2̄

∣∣∣+
∣∣∣− 2̄−

(
−5̄
2̄

)∣∣∣ = 1̄
2̄

+ 1̄
2̄

= 1̄.



REFERENCES 956

The solution shows that the shape of spheres undergoes radical changes as the definition
of soft metric changes.

5. Conclusion

This article is intended to revive soft spheres with soft real numbers and soft points
in soft metric spaces of curvature bounded below and a survey of applicable results. The
locus of the soft sphere is analyzed and the most important results are proved compre-
hensively. I hope that this article will provide a good beginning point for mathematicians
interested in this area and a common reference for future papers on this subject.

Acknowledgements

The author would like to thank the anonymous referee for invaluable suggestions which
helped in improving the contents of this manuscript.

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