@ Appl. Gen. Topol. 22, no. 2 (2021), 447-459doi:10.4995/agt.2021.15566
© AGT, UPV, 2021

Index boundedness and uniform connectedness

of space of the G-permutation degree

R. B. Beshimov a, D. N. Georgiou b and R. M. Zhuraev a

a
National University of Uzbekistan named after Mirzo Ulugbek, str. University, 100174 Tashkent,

Uzbeksitan (rbeshimov@mail.ru,rmjurayev@mail.ru)
b
University of Patras, Department of mathematics, 26504 Patras, Greece (georgiou@math.upatras.gr)

Communicated by S. Romaguera

Abstract

In this paper the properties of space of the G-permutation degree, like:
weight, uniform connectedness and index boundedness are studied. It
is proved that:
(1) If (X, U) is a uniform space, then the mapping πsn, G : (X

n
, U

n) →
(SP nGX, SP

n
GU) is uniformly continuous and uniformly open, moreover

w (U) = w (SP nGU);
(2) If the mapping f : (X, U) → (Y, V) is a uniformly continuous
(open), then the mapping SP nGf : (SP

n
GX, SP

n
GU) → (SP

n
GY, SP

n
GV)

is also uniformly continuous (open);
(3) If the uniform space (X, U) is uniformly connected, then the uni-
form space (SP nGX, SP

n
GU) is also uniformly connected.

2010 MSC: 54A05; 54E15; 55S15.

Keywords: G-permutation degree space; uniform space; uniform connected-
ness; index boundedness of uniform space; uniform continuity.

1. Introduction

In [19], a functor O : Comp → Comp of weakly additive functionals acting
in the category of compact and its continuous mappings is defined. It was
proved that the functor O : Comp → Comp satisfies the normality conditions,
except the preimage preservation condition. In [6], categorical and cardinal

Received 05 May 2021 – Accepted 08 June 2021

http://dx.doi.org/10.4995/agt.2021.15566


R. B. Beshimov, D. N. Georgiou and R. M. Zhuraev

properties of hyperspaces with finite number of components are investigated.
It was proved that the functor Cn : Comp → Comp is not normal, i.e., it
does not preserve epimorphisms of continuous mappings. The authors of this
paper also discussed the properties of density, caliber and Shanin number for
the space Cn(X). This space Cn(X) is of great interest for researchers, since
it contains the hyperspaces expn X of closed sets with cardinalities not greater
than n elements. It was proved in [4] that the Radon functor satisfies all the
normality conditions.

In [7], the topological properties of topological groups were studied. In
[5], categorical and topological properties of the functor OSτ of semiadditive
τ-smooth functionals in the category T ych of Tychonoff spaces and their con-
tinuous mappings, which extends the functors OS of semiadditive functionals
in the category Comp of compact and their continuous mappings, were inves-
tigated.

In [3], some properties of the functor Oβ : T ych → T ych were considered,
where β is the Čech-Stone compact extension in the category of Tychonoff
spaces and their continuous mappings. This functor is regarded as an extension
of the functor O : Comp → Comp. The author in [3] proved that the space
Oβ(X) is a convex subset of the space Cp(Cb(X)), where

Cb(X) = {f ∈ C(X)| f : X → R is a bounded function}
and Cp(X) is the space of pointwise convergence. It was proved in [2] that if
a covariant functor F : Comp → Comp is weakly normal, then Fβ : T ych →
T ych does not increase the density and weak density for any infinite Tychonoff
space.

In [11], it was proved that the functor SP nG preserves the property of the
fibers of the map to be a compact Q-manifold. In [9] some classes of uniform
spaces are considered. In particular, the uniformly continuous mappings and
absolutes, generalizations of metrics, normed, uniform unitary spaces, topo-
logical and uniform groups, its completions and spectral characterizations are
studied. In addition, the properties of uniformly continuous and uniformly
open mappings between uniform spaces are studied, too. But, it should be
noted here that the class of uniformly continuous and uniformly open maps
itself was introduced by Michael in [18].

In what follows, we present the basic notions that will be used in the rest of
this article.

It is known that a permutation group is the group of all permutations, that
is one-to-one mappings X → X. A permutation group of a set X is usually
denoted by S(X). Especially, if X = {1, 2, . . . , n}, then S(X) is denoted by
Sn.

Let Xn be the n-th power of a compact space X. The permutation group
Sn of all permutations acts on the n-th power X

n as permutation of coordi-
nates. The set of all orbits of this action with quotient topology is denoted
by SP nX. Thus, points of the space SP nX are finite subsets (equivalence
classes) of the product Xn. Two points (x1, x2, . . . , xn), (y1, y2, . . . , yn) ∈ Xn

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 448



Index boundedness and uniform connectedness of space of the G-permutation degree

are considered to be equivalent if there exists a permutation σ ∈ Sn such that
yi = xσ(i). The space SP

nX is called the n-permutation degree of the space
X. Equivalent relation by which we obtain space SP nX is called the sym-
metric equivalence relation. The n-th permutation degree is a quotient of Xn.
Therefore, the quotient map is denoted by πsn : X

n → SP nX, where for every
x = (x1, x2, . . . , xn) ∈ Xn,

πsn((x1, x2, . . . , xn)) = [(x1, x2, . . . , xn)]

is an orbit of the point x = (x1, x2, . . . , xn) ∈ Xn [20].
Let G be a subgroup of the permutation group Sn and let X be a compact

space. The group G acts also on the n-th power of the space X as permutation
of coordinates. The set of all orbits of this action with quotient topology is
denoted by SP nGX. The space SP

n
GX is called G-permutation degree of the

space X [13]. Actually, it is the quotient space of the product of Xn under the
G-symmetric equivalence relation.

An operation SP n is the covariant functor in the category of compacts and
it is said to be a functor of G-permutation degree. If G = Sn, then SP

n
G = SP

n

and if the group G consists of the unique element only, then SP n = Xn.

Let T be a set and let A and B be subsets of T × T , i.e., relations on the
set T . The inverse relation of A will be denoted by A−1, that is,

A−1 = {(x, y) : (y, x) ∈ A}.
The composition of A and B will be denoted by AB; thus we have

AB = {(x, z) : there exists a y ∈ T such that (x, y) ∈ A and (y, z) ∈ B}.
For an arbitrary relation A ⊂ T × T and for a positive integer n the relation
An ⊂ T × T is defined inductively by the formulas:

A1 = A and An = An−1A.

Every set V ⊂ T × T that contains the diagonal ∆T = {(x, x) : x ∈ T } of T is
called an entourage of the diagonal.

Definition 1.1. Let T be a non-empty set. A family U of subsets of T × T is
called a uniformity on T , if this family satisfies the following conditions:

(U1) Each U ∈ U contains the diagonal ∆T = {(x, x) : x ∈ T } of T ;
(U2) If V1, V2 ∈ U, then V1 ∩ V2 ∈ U;
(U3) If U ∈ U and U ⊂ V , then V ∈ U;
(U4) For each U ∈ U there is a V ∈ U such that V 2 ⊂ U;
(U5) For each U ∈ U we have U−1 ∈ U.
The pair (T, U) is called uniform space [17]. Also, the elements of the uni-

formity U are called entourages. For an entourage U ∈ U and a point x ∈ T
the set

U(x) = {y ∈ T : (x, y) ∈ U}
is called the U-ball centered at x. For a subset A ⊂ T the set U(A) =

⋃

a∈A

U(a)

is called the U-neighborhood of A [1].

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 449



R. B. Beshimov, D. N. Georgiou and R. M. Zhuraev

A family B is called a base for the uniformity U, if for any V ∈ U there
exists a W ∈ B with W ⊂ U. The smallest cardinal number of the form |B|,
where B is a base for U, is called the weight of the uniformity U and is denoted
by ω(U).

Every base B for a uniformity on T has the following properties:
(BU1) For every V1, V2 ∈ B there exists a V ∈ B such that V ⊂ V1 ∩ V2;
(BU2) For every V ∈ B there exists a W ∈ B such that W 2 ⊂ V .

Proposition 1.2 ([15]). Suppose that a non-empty set X is given. Consider a
family B of entourages of the diagonal, which has the properties (BU1)–(BU2)
and B−1 = B. A family U is a uniformity on X, if it consists of all entourages
which contain a member of B. The family B is a base for U. The uniformity
U is called the uniformity generated by the base B.

Let {(Xs, Us) : s ∈ S} be a family of uniform spaces. A family B of all
entourages of the diagonal, which has the form

{({xs} , {ys}) : (xsi , ysi) ∈ Vsi for s1, s2, . . . , sk ∈ S, Vsi ∈ Usi, i = 1, 2, . . . , k} ,
generates a uniformity on the set

∏

s∈S

Xs. This uniformity is called a Cartesian

product of the uniformities {Us : s ∈ S} and is denoted by
∏

s∈S

Us. If all the
uniformities Us are equal to each other, i.e., if Xs = X and Us = U for s ∈ S,
then the Cartesian product

∏

s∈S

Us is also denoted by Uτ, where τ = |S| [10].

Definition 1.3. A function f : (X, U) → (Y, V) is called uniformly continuous,
if for each V ∈ V there exists a U ∈ U such that

(f × f)(U) = {(f(x1), f(x2)) : (x1, x2) ∈ U} ⊂ V.
Note that the condition (f × f)(U) ⊂ V is equivalent to the condition

f(U(x)) ⊂ V (f(x)) or (f × f)−1(V ) ∈ U [14].
A uniformly continuous function f : (X, U) → (Y, V) is uniformly open, if

for any U ∈ U there exists a V ∈ V such that
V (f(x)) ⊂ f(U(x))

for all x ∈ X [12].
Let expc X and expc Y be the hyperspaces of X and Y , consisting of all

nonempty compact subsets equipped with the Hausdorff uniformity. In [12], it
was proved that if a (continuous) surjection f : X → Y between uniform spaces
X and Y is perfect, then f is uniformly open if and only if expc f : expc X →
expc Y is uniformly open.

Remark 1.4. The uniform continuity of the mapping f does not always imply
uniform openness, i.e., there is a mapping f that can be uniformly continuous,
but cannot be uniformly open.

As an example, consider the mapping f : (R, U) → (R, V), where f(x) = x,
x ∈ X, V = {∆, R × R}, U = {U ⊂ R × R : ∆ = {(x, x) : x ∈ R} ⊂ U} and

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 450



Index boundedness and uniform connectedness of space of the G-permutation degree

R is the set of all real numbers. This mapping f is uniformly continuous, but
not uniformly open.

Recall that a bijective mapping f : (X, U) → (Y, V), acting from the uniform
space (X, U) to the uniform space (Y, V), is called a uniform isomorphism if
the mappings f : (X, U) → (Y, V) and f−1 : (Y, V) → (X, U) are uniformly
continuous [8].

Let (X, U) be a uniform space and D ∈ U. A pair of points x, y of the
uniform space (X, U) is said to be related by a D-chain, if there exists an
integer k such that (x, y) ∈ Dk. The uniform space X is called uniformly
connected, if every entourage D of X and every pair of points of X are related
by a D-chain [16].

In our paper we use the following theorem, which have been proved in [15].

Theorem 1.5. The uniformly continuous image of a uniformly connected space
is uniformly connected.

The smallest cardinal number τ is called an index boundedness of a uniform
space (X, U), if the uniformity U has a base B consisting of entourages of
cardinality ≤ τ. The index boundedness is denoted by l(U). The uniform
space (X, U) is called τ-bounded, if l(U) ≤ τ [8].

2. Uniformly open and uniformly continuous mappings

Theorem 2.1. Let (X, U) be a uniform space. A family B of all subsets of
SP nGX × SP nGX of the form

O
[

U1, U2, . . . , Un
]

= {([x], [y]) : there exist permutations
σ, δ ∈ G such that

(

xi, yσ(i)
)

∈ Uδ(i), i = 1, 2, . . . , n},
where

{

U1, U2, . . . , Un
}

⊂ U, has the properties (BU1)–(BU2) and generates
some uniformity on SP nGX.

Proof. First, we show that every set of the form O
[

U1, U2, . . . , Un
]

is an en-

tourage of the diagonal, where
{

U1, U2, . . . , Un
}

⊂ U. Take an arbitrary point
[x] = [(x1, x2, . . . , xn)] ∈ SP nGX. Then

(

xi, xi
)

∈ Ui for all i = 1, 2, . . . , n. In
this case, we have that σ = δ = e is the unit element of the group G. Therefore,

([x], [x]) ∈ O
[

U1, U2, . . . , Un
]

and
∆ = {([x], [x]) : [x] ∈ SP nGX} ⊂ O

[

U1, U2, . . . , Un
]

.

Choose any two entourages O
[

U1, U2, . . . , Un
]

and O
[

V1, V2, . . . , Vn
]

of the
family B. It is clear, that Ui ∩ Vi ∈ U for each i = 1, 2, . . . , n. So, it is enough
to show the following relation:

O
[

U1 ∩ V1, U2 ∩ V2, . . . , Un ∩ Vn
]

⊂ O
[

U1, U2, . . . , Un
]

∩ O
[

V1, V2, . . . , Vn
]

.

Let
([x], [y]) ∈ O

[

U1 ∩ V1, U2 ∩ V2, . . . , Un ∩ Vn
]

.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 451



R. B. Beshimov, D. N. Georgiou and R. M. Zhuraev

Then there exist permutations θ, φ ∈ G such that
(

xi, yθ(i)
)

∈
(

Uφ(i) ∩ Vφ(i)
)

for all i = 1, 2, . . . , n. Put
(

xi, yθ(i)
)

∈ Uφ(i) and
(

xi, yθ(i)
)

∈ Vφ(i) for all
i = 1, 2, . . . , n. It means that

([x], [y]) ∈ O
[

U1, U2, . . . , Un
]

∩ O
[

V1, V2, . . . , Vn
]

.

For any entourage O
[

U1, U2, . . . , Un
]

∈ B there is a Wi ∈ U such that
W 2i ⊂ Ui for each i = 1, 2, . . . , n. Put W =

n
⋂

i=1

Wi. We prove that

O
[

W ′1, W
′
2, . . . , W

′
n

]2 ⊂ O
[

U1, U2, . . . , Un
]

,

where W ′i = W for every i = 1, 2, . . . , n.
Let

([x], [y]) ∈ O
[

W ′1, W
′
2, . . . , W

′
n

]2
.

Then there exists an orbit [z] ∈ SP nGX such that ([x], [z]) ∈ O
[

W ′1, W
′
2, . . . , W

′
n

]

and ([z], [y]) ∈ O
[

W ′1, W
′
2, . . . , W

′
n

]

. Since ([x], [z]) ∈ O
[

W ′1, W
′
2, . . . , W

′
n

]

there are permutations σ1, δ1 ∈ G such that
(

xi, zσ1(i)
)

∈ W ′
δ1(i)

= W for

all i = 1, 2, . . . , n. If ([z], [y]) ∈ O
[

W ′1, W
′
2, . . . , W

′
n

]

, then there exist permu-

tations ϕ, γ ∈ G such that
(

zj, yϕ(j)
)

∈ W ′
γ(j)

= W for all j = 1, 2, . . . , n.

Put j = σ1(i) and we obtain
(

xi, zσ1(i)
)

∈ W and
(

zσ1(i), yϕσ1(i)
)

∈ W .
Thus,

(

xi, yϕσ1(i)
)

∈ W 2 ⊂ W 2i ⊂ Ui for each i = 1, 2, . . . , n. Consequently
([x], [y]) ∈ O[U1, U2, . . . , Un], i.e. O[W ′1, W ′2, . . . , W ′n]2 ⊂ O

[

U1, U2, . . . , Un
]

.
Now we prove that

O
[

U1, U2, . . . , Un
]−1

= O
[

U−11 , U
−1
2 , . . . , U

−1
n

]

.

Indeed, let ([x], [y]) ∈ O
[

U1, U2, . . . , Un
]−1

. Then ([y], [x]) ∈ O
[

U1, U2, . . . , Un
]

and there are permutations σ2, δ2 ∈ G such that
(

yi, xσ2(i)
)

∈ Uδ2(i) for ev-
ery i = 1, 2, . . . , n. Put j = σ2(i). This implies that i = σ

−1
2 (j). The

relation
(

y
σ
−1

2
(j), xj

)

∈ U
δ2σ

−1

2
(j) implies that

(

xj, yσ−1
2

(j)

)

∈ U−1
δ2σ

−1

2
(j)

for

all j = 1, 2, . . . , n. Therefore, ([x], [y]) ∈ O
[

U−11 , U
−1
2 , . . . , U

−1
n

]

. We have

O
[

U1, U2, . . . , Un
]−1 ⊂ O

[

U−11 , U
−1
2 , . . . , U

−1
n

]

. The reverse inclusion is simi-
larly.

By Proposition 1.2, the family B generates some uniformity SP nGU on the
set SP nGX. Theorem 2.1 is proved. �

Consider a mapping πsn, G : (X
n, Un) →

(

SP nGX, SP
n
GU

)

defining as follows:

πsn, G
(

x1, x2, . . . , xn
)

=
[

(x1, x2, . . . , xn)
]

G

for each
(

x1, x2, . . . , xn
)

∈ Xn.
Theorem 2.2. Let (X, U) be a uniform space. Then the mapping

πsn, G :
(

Xn, Un) → (SP nGX, SP nGU
)

is uniformly continuous.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 452



Index boundedness and uniform connectedness of space of the G-permutation degree

Proof. Let O
[

U1, U2, . . . , Un
]

be any entourage in
(

SP nGX, SP
n
GU

)

. Consider
an entourage

U =
{

(a, b) :
(

ai, bi
)

∈ Ui, i = 1, 2, . . . , n
}

in
(

Xn, Un
)

, where a =
(

a1, a2, . . . , an
)

and b =
(

b1, b2, . . . , bn
)

are points of

Xn. We prove that for all x =
(

x1, x2, . . . , xn
)

∈ Xn,
πsn, G(U(x)) ⊂ O

[

U1, U2, . . . , Un
]

([x]).

Indeed, if y =
(

y1, y2, . . . , yn
)

∈ U(x), then
(

xi, yi
)

∈ Ui for each i =
1, 2, . . . , n. From

(

xi, yi
)

∈ Ui we have
(

xi, yσ(i)
)

∈ Uδ(i), where σ = δ =
e ∈ G. In this case, we have that [y] ∈ O

[

U1, U2, . . . , Un
]

([x]). Thus,

πsn, G(U(x)) ⊂ O
[

U1, U2, . . . , Un
]

([x]). Theorem 2.2 is proved. �

Theorem 2.3. For a uniform space (X, U) the mapping
πsn, G :

(

Xn, Un
)

→
(

SP nGX, SP
n
GU

)

is uniformly open.

Proof. By Theorem 2.2 the mapping πsn, G is uniformly continuous. Let U be

an arbitrary entourage in Un. Then there is a trace {U1, U2, . . . , Un} ⊂ U such
that

{

(a, b) :
(

ai, bi
)

∈ Ui, i = 1, 2, . . . , n
}

⊂ U,
where a =

(

a1, a2, . . . , an
)

and b =
(

b1, b2, . . . , bn
)

are points of Xn. We show

that for any point x =
(

x1, x2, . . . , xn
)

∈ Xn we have
O
[

U′1, U
′
2, . . . , U

′
n

]

([x]) ⊂ πsn, G(U(x)),

where U′k =
n
⋂

i=1

Ui for k = 1, 2, . . . , n.

Indeed, if [y] ∈ O
[

U′1, U
′
2, . . . , U

′
n

]

([x]), then there exist permutations σ, δ ∈
G such that

(

xi, yσ(i)
)

∈ U′
δ(i)

for all i = 1, 2, . . . , n. In particular,
(

xi, yσ(i)
)

∈
Ui for all i = 1, 2, . . . , n, i.e.,

(2.1) (x, yσ) ∈
{

(a, b) :
(

ai, bi
)

∈ Ui, i = 1, 2, . . . , n
}

where yσ =
(

yσ(1), yσ(2), ..., yσ(n)
)

. From (2.1) it follows that yσ ∈ U(x) and
[y] ∈ πsn, G(U(x)). Therefore, O

[

U′1, U
′
2, . . . , U

′
n

]

([x]) ⊂ πsn, G(U(x)) for a point
x ∈ Xn. Theorem 2.3 is proved. �

Proposition 2.4. Let f : (X, U) → (Y, V) be a uniformly open mapping and
f(X) = Y . Then w(V) ≤ w(U).
Proof. Let w(U) = τ ≥ ℵ0. Then there is a base B = {Uα : α ∈ M} of
uniformity U such that |M| = τ. We shall prove that the family

(f × f)(B) = {(f × f)(Uα) : α ∈ M}
is a base of uniformity V. Since the map f is uniformly open, we have that
(f × f)(Uα) ∈ V for each α ∈ M. For any entourage V ∈ V the relation
(f × f)−1(V ) ∈ U is true. In this case, there exists an index α0 ∈ M such

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 453



R. B. Beshimov, D. N. Georgiou and R. M. Zhuraev

that Uα0 ⊂ (f × f)−1(V ), i.e. (f × f)(Uα0) ⊂ V . It means that the family
(f × f)(B) is a base of uniformity V. Proposition 2.4 is proved. �

For a uniform space (X, U) we define a mapping λ : X → SP nGX, where
λ(x) = [(x, x, . . . , x)], x ∈ X.
Proposition 2.5. For a uniform space (X, U) the mapping

λ : (X, U) →
(

SP nGX, SP
n
GU

)

is a uniform embedding.

Proof. Let λ|∆ : X → ∆ be the restriction of the map λ : X → SP nGX, where
∆ = {[(x, x, . . . , x)] : x ∈ X}. It is known that it is bijective. Let us show
that the map λ|∆ is uniformly continuous. Choose an arbitrary entourage
O
[

U1, U2, . . . , Un
]

∈ SP nGU. Put U =
n
⋂

i=1

Ui. By the definition of uniformity

we have U ∈ U. It suffices to show that
λ|∆(U(x)) ⊂

(

O
[

U1, U2, . . . , Un
]

∩ (∆ × ∆)
)(

λ|∆(x)
)

for all x ∈ X. Clearly, λ|∆(x) = λ(x) and
(

O
[

U1, U2, . . . , Un
]

∩ (∆ × ∆)
)

(λ|∆(x)) = O
[

U1, U2, . . . , Un
]

(λ(x)) ∩ ∆
for x ∈ X. Let y ∈ U(x). Then (x, y) ∈ U ⊂ Ui for every i = 1, 2, . . . , n. In
this case, we have (λ(x), λ(y)) ∈ O

[

U1, U2, . . . , Un
]

, i.e.,

λ(y) ∈ O
[

U1, U2, . . . , Un
]

(λ(x)) ∩ ∆.
Now we show that the mapping (λ|∆)−1 : ∆ → X is uniformly continuous.

Take an arbitrary entourage V ∈ U. The following relation holds:
(λ|∆)−1

(

O
[

V ′1, V
′
2, . . . , V

′
n

]

(λ(x)) ∩ ∆
)

⊂ V (x), x ∈ X,

where V ′k =
n
⋂

i=1

Vi for each k = 1, 2, . . . , n. It means that the mapping (λ|∆)−1

is uniformly continuous. Proposition 2.5 is proved. �

Lemma 2.6 ([8]). Let (X, U) be a uniform space and (Y, U|Y ) be its subspace,
where U|Y = {U ∩ (Y × Y ) : U ∈ U}. Then w(U|Y ) ≤ w(U).
Theorem 2.7. Let (X, U) be a uniform space. Then the equality w(U) =
w(SP nGU) holds.
Proof. Let (X, U) be a uniform space. By Proposition 2.5 and Lemma 2.6 it
follows that w(U) ≤ w(SP nGU). By the definition of uniformity SP nGU on the set
SP nGX we have w(SP

n
GU) ≤ w(U). Thus, we directly obtain w(U) = w(SP nGU).

Theorem 2.7 is proved. �

Consider an arbitrary mapping f : (X, U) → (Y, V), where (X, U) and
(Y, V) are uniform spaces. For an equivalence class [(x1, x2, . . . , xn)] ∈ SP nGX,
put

SP nGf
[(

x1, x2, . . . , xn
)]

G
=

[(

f
(

x1
)

, f
(

x2
)

, . . . , f
(

xn
))]

G
.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 454



Index boundedness and uniform connectedness of space of the G-permutation degree

The following mapping is defined

SP nGf :
(

SP nGX, SP
n
GU

)

→
(

SP nGY, SP
n
GV

)

.

We obtained the following result.

Theorem 2.8. Let f : (X, U) → (Y, V) be a uniformly continuous mapping.
Then the mapping SP nGf :

(

SP nGX, SP
n
GU

)

→
(

SP nGY, SP
n
GV

)

is also uni-
formly continuous.

Proof. Let f : (X, U) → (Y, V) be a uniformly continuous mapping. Take an
arbitrary entourage O

[

V1, V2, . . . , Vn
]

∈ SP nGV. Then there is an entourage
Ui ∈ U such that f

(

Ui(a)
)

⊂ Vi
(

f(a)
)

for all a ∈ X and i = 1, 2, . . . , n. We
show that

SP nGf
(

O
[

U1, U2, . . . , Un
]

([x])
)

⊂ O
[

V1, V2, . . . , Vn
](

SP nGf([x])
)

for a point [x] ∈ SP nGX. Choose an orbit [y] ∈ O
[

U1, U2, . . . , Un
]

([x]). Then

there exist permutations σ, δ ∈ G such that
(

xi, yσ(i)
)

∈ Uδ(i) for all i =
1, 2, . . . , n. We have yσ(i) ∈ Uδ(i)

(

xi
)

for any i = 1, 2, . . . , n. Therefore,

f
(

yσ(i)
)

∈ f(Uδ(i)
(

xi)
)

⊂ Vδ(i)
(

f
(

xi
))

.

It means that

(2.2)
(

f
(

xi
)

, f
(

yσ(i)
))

∈ Vδ(i)
for all i = 1, 2, . . . , n. Put

SP nGf([x]) =
[(

f
(

x1
)

, f
(

x2
)

, . . . , f
(

xn
))]

G

and

SP nGf([y]) =
[(

f(y1
)

, f
(

y2
)

, . . . , f
(

yn
))]

G
.

From (2.2) we obtain
(

SP nGf([x]), SP
n
Gf([y])

)

∈ O
[

V1, V2, . . . , Vn
]

.

Hence, SP nGf([y]) ∈ O
[

V1, V2, . . . , Vn
](

SP nGf([x])
)

. Theorem 2.8 is proved.
�

Theorem 2.9. Let f : (X, U) → (Y, V) be a uniformly open mapping. Then
the mapping SP nGf :

(

SP nGX, SP
n
GU

)

→
(

SP nGY, SP
n
GV

)

is also uniformly
open.

Proof. Let f : (X, U) → (Y, V) be a uniformly open mapping. Take an arbi-
trary entourage O

[

U1, U2, . . . , Un
]

∈ SP nGU. In this case there exists entourage
Vi ∈ V such that Vi(f(a)) ⊂ f(Ui(a)) for each a ∈ X and i = 1, 2, . . . , n. It
suffices to show that

O
[

V1, V2, . . . , Vn
](

SP nGf([x])
)

⊂ SP nGf
(

O
[

U1, U2, . . . , Un
]

([x])
)

.

Choose an arbitrary point [y] ∈ O
[

V1, V2, . . . , Vn
](

SP nGf([x])
)

. Then there are

permutations σ, δ ∈ G such that
(

f
(

xi
)

, yσ(i)
)

∈ Vδ(i) for all i = 1, 2, . . . , n.
Moreover, yσ(i) ∈ Vδ(i)(f(xi)) ⊂ f(Uδ(i)(xi)) for each i = 1, 2, . . . , n. Since

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 455



R. B. Beshimov, D. N. Georgiou and R. M. Zhuraev

yσ(i) ∈ f
(

Uδ(i)
(

xi
))

, there exists a point zi ∈ Uδ(i)
(

xi
)

such that yσ(i) = f
(

zi
)

for all i = 1, 2, . . . , n. Put z =
(

z1, z2, . . . , zn
)

∈ Xn and we have
(2.3) [y] = SP nGf([z]).

The relation zi ∈ Uδ(i)
(

xi
)

implies

(2.4) [z] ∈ O
[

U1, U2, . . . , Un
]

([x])

for i = 1, 2, . . . , n. By relations (2.3) and (2.4) it follows that

[y] ∈ SP nGf
(

O
[

U1, U2, . . . , Un
]

([x])
)

.

Hence

O
[

V1, V2, . . . , Vn
](

SP nGf([x])
)

⊂ SP nGf
(

O
[

U1, U2, . . . , Un
]

([x])
)

.

Theorem 2.9 is proved. �

3. Uniformly connected spaces and index boundedness

Theorem 3.1. If a uniform space (X, U) is uniformly connected, then the
uniform space

(

SP nGX, SP
n
GU

)

is also uniformly connected.

Proof. Let x, y ∈ Xn, where x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn).
Take an arbitrary entourage

U = {(a, b) : (ai, bi) ∈ Ui, i = 1, 2, . . . , n} ∈ Un

on Xn, where a = (a1, a2, . . . , an) and b = (b1, b2, ..., bn). Since X is uniformly

connected, there is a ki ∈ Z such that (xi, yi) ∈ Ukii for every i = 1, 2, . . . , n.
Put k = max{ki : i = 1, 2, . . . , n}. Therefore, (xi, yi) ∈ Uki for each i =
1, 2, . . . , n. In this case there are points z1i , z

2
i , . . . , z

k−1
i such that















(

xi, z
1
i

)

∈ Ui,
(

z1i , z
2
i

)

∈ Ui,
. . . .

(

zk−1i , yi
)

∈ Ui.
for every i = 1, 2, ..., n. Consider k − 1 points of Xn;
z1 =

(

z11, z
1
2, . . . , z

1
n

)

, z2 =
(

z21, z
2
2, . . . , z

2
n

)

, . . . , zk−1 =
(

zk−11 , z
k−1
2 , . . . , z

k−1
n

)

.

By definition of entourage U, we have
(

x, z1
)

∈ U,
(

z1, z2
)

∈ U, . . . ,
(

zk−1, y
)

∈ U.
It means that (x, y) ∈ Uk. Hence,

(

Xn, Un
)

is uniformly connected space.

By Theorem 1.1 [15] and Theorem 2.2 the space
(

SP nGX, SP
n
GU

)

is uniformly
connected. Theorem 3.1 is proved. �

We say that a uniform space (X, U) is discrete, if ∆X ∈ U [15].
Example 3.2. Any discrete uniform space is not uniformly connected.

Indeed, take points x, y ∈ X with x 6= y. Then for any integer number k we
have (x, y) /∈ ∆X = ∆kX.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 456



Index boundedness and uniform connectedness of space of the G-permutation degree

Theorem 3.3 ([15]). For every uniformity U on a set X the family

τU = {G ⊂ X : for every x ∈ G there exists a V ∈ U such that V (x) ⊂ G}
is a topology on the set X, which is called the topology induced by the uniformity
U.
Remark 3.4. The uniformly connectedness of the space (X, U) does not imply
connectedness with respect to the topology induced by the uniformity U, in
general.

Consider the family B = {Uε : ε > 0} of subsets of R × R, where Uε =
{(x, y) ∈ R × R : |x − y| < ε} and R is the real line. The family B has
the properties (BU1)–(BU2) and generates some uniformity on the real line R.
This uniformity is called natural uniformity on R.

The family BQ = {Uε ∩ (Q × Q) : ε > 0} is a base of a uniformity on Q (the
set of all rational numbers) and generates some uniformity UQ on Q. For any
ε > 0 and r1, r2 ∈ Q (r1 < r2) we have k = [r2−r1ε ] + 1. Consider a sequence of
points {ai}k−1i=1 defined by the formula ai = r1 +

r2−r1
k

i with i = 1, 2, . . . , k − 1.
It is clear that ai ∈ Q for all i = 1, 2, . . . , k − 1. In this case we have















(

r1, a1
)

∈ Vε,
(

a1, a2
)

∈ Vε,
. . . .

(

ak−1, r2
)

∈ Vε,
where Vε = Uε ∩ (Q × Q). Thus,

(

r1, r2
)

∈ V kε . Therefore, the uniform space
(

Q, UQ
)

is uniformly connected, but not connected, since Q = ((−∞,
√
2) ∩

Q) ∪ ((
√
2, ∞) ∩ Q) and ((−∞,

√
2) ∩ Q) ∩ ((

√
2, ∞) ∩ Q) = ∅.

Theorem 3.5. Let (X, U) be a uniform space. Then the equality l(U) =
l
(

SP nGU
)

holds.

Proof. First, we show the inequality l
(

SP nGU
)

≤ l
(

Un
)

. Let l
(

Un
)

= τ ≥ ℵ0.
Then there is a base Bn of uniformity Un such that |V | ≤ τ for any V ∈ Bn. We
consider the family πsn, G

(

Bn
)

= {πsn, G(V ) : V ∈ Bn} and prove that the family
πsn, G

(

Bn
)

is the base of the uniformity SP nGU. Since πsn, G is a uniformly open
mapping, πsn, G(V ) ∈ SP nGU for any entourage V ∈ Bn. Consider an arbitrary
entourage

O
[

U1, U2, . . . , Un
]

∈ SP nGU.
For an entourage

(

πsn, G
)−1(

O
[

U1, U2, . . . , Un
])

∈ Un there exists an entourage
V ∈ Bn such that

V ⊂
(

πsn, G
)−1(

O
[

U1, U2, . . . , Un
])

.

Hence, we obtain
πsn, G(V ) ⊂ O

[

U1, U2, . . . , Un
]

,

i.e., we have that l
(

SP nGU
)

≤ τ.
Now we show the inverse inequality l

(

Un
)

≤ l(U). Let l(U) = κ ≥ ℵ0 and let
B be a base for the uniformity U, consisting of entourages of cardinality ≤ κ.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 457



R. B. Beshimov, D. N. Georgiou and R. M. Zhuraev

We denote by B′ the family of all entourages of the form
n
⋂

i=1

pr−1i
(

Ui
)

, where

Ui ∈ B and pri is the projection of Xn onto Xi = X for each i = 1, 2, . . . , n.
Then by definition of Cartesian product of the uniform spaces, the family B′ is
a base for the uniformity Un. Since

|
n
⋂

i=1

pr−1i (Ui)| ≤ max{|Ui| : i = 1, 2, . . . , n} ≤ κ

it follows that l
(

Un
)

≤ l(U).
Consequently, we have l

(

SP nGU
)

≤ l(U). Note that a uniform subspace of
a τ-bounded space is also τ-bounded. By Proposition 2.5, l(U) ≤ l(SP nGU).
Hence, we get the equality l(U) = l

(

SP nGU
)

. �

References

[1] T. Banakh, Topological spaces with an ωω-base, Dissertationes Mathematicae,
Warszawa, 2019.

[2] R. B. Beshimov, Nonincrease of density and weak density under weakly normal functors,
Mathematical Notes 84 (2008), 493–497.

[3] R. B. Beshimov, Some properties of the functor Oβ, Journal of Mathematical Sciences
133, no. 5 (2006), 1599–1601.

[4] R. B. Beshimov and N. K. Mamadaliev, Categorical and topological properties of the
functor of Radon functionals, Topology and its Applications 275 (2020), 1–11.

[5] R. B. Beshimov and N. K. Mamadaliev, On the functor of semiadditive τ-smooth func-
tionals, Topology and its Applications 221, no. 3 (2017), 167–177.

[6] R. B. Beshimov, N. K. Mamadaliev, Sh. Kh. Eshtemirova, Categorical and cardinal
properties of hyperspaces with a finite number of components, Journal of Mathematical
Sciences 245, no. 3 (2020), 390–397.

[7] R. B. Beshimov and R. M. Zhuraev, Some properties of a connected topological group,
Mathematics and Statistics 7, no. 2 (2019), 45–49.

[8] A. A. Borubaev and A. A. Chekeev, On completions of topological groups with respect
to the maximal uniform structure and factorization of uniform homomorphisms with
respect to uniform weight and dimension, Topology and its Applications 107, no. 1-2
(2000), 25–37.

[9] A. A. Borubaev and A. A. Chekeev, On uniform topology and its applications, TWMS
J. Pure and Appl. Math. 6, no. 2 (2015), 165–179.

[10] R. Engelking, General topology, Berlin: Helderman, 1986.
[11] V. V. Fedorchuk, Covariant functors in the category of compacts, absolute retracts and

Q-manifolds, Uspekhi Matematicheskikh Nauk 36, no. 3 (1981), 177–195.
[12] V. V. Fedorchuk and H. A. Kunzi, Uniformly open mappings and uniform embeddings

of function spaces, Topology and its Applications 61 (1995), 61–84.
[13] V. V. Fedorchuk and V. V. Filippov, Topology of hyperspaces and its applications, 4

Mathematica, cybernetica, Moscow, 48 p., 1989 (in Russian).
[14] G. Itzkowitz, S. Rothman, H. Strassberg and T. S. Wu, Characterization of equivalent

uniformities in topological groups, Topology and its Applications 47 (1992), 9–34.
[15] I. M. James, Introduction to Uniform Spaces, London Mathematical Society, Lecture

Notes Series 144, Cambridge University Press, Cambridge, 1990.
[16] J. L. Kelley, General Topology, Van Nostrand Reinhold, Princeton, NJ, 1955.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 458



Index boundedness and uniform connectedness of space of the G-permutation degree

[17] L. Holá and L. D. R. Kocinac, Uniform boundedness in function spaces, Topology and
its Applications 241 (2018), 242–251.

[18] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182.
[19] T. N. Radul, On the functor of order-preserving functionals, Comment. Math. Univ.

Carol. 39, no. 3 (1998), 609–615.
[20] T. K. Yuldashev and F. G. Mukhamadiev, The local density and the local weak density

in the space of permutation degree and in Hattorri space, URAL Mathematical Journal
6, no. 2 (2020), 108–126.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 459