@ Appl. Gen. Topol. 22, no. 2 (2021), 345-354doi:10.4995/agt.2021.14809
© AGT, UPV, 2021

Sum connectedness in proximity spaces

Beenu Singh
a
and Davinder Singh

b

a
Department of Mathematics, University of Delhi, Delhi (India) - 110007. (singhbeenu47@gmail.com)

b
Department of Mathematics, Sri Aurobindo College (M), University of Delhi, Delhi (India) -

110017. (dstopology@gmail.com)

Communicated by P. Das

Abstract

The notion of sum δ-connected proximity spaces which contain the

category of δ-connected and locally δ-connected spaces is defined. Sev-

eral characterizations of it are substantiated. Weaker forms of sum

δ-connectedness are also studied.

2010 MSC: 54E05; 54D05.

Keywords: sum δ-connected; δ-connected; δ-component; locally δ-

connected.

1. Introduction

The notion of proximity was introduced by Efremovic [4, 5] as a natural gen-
eralization of metric spaces and topological groups. Smirnov [10, 11] and Naim-
pally [8, 9] did the most significant and extensive work in this area. In 2009,
Bezhanishvili [1] defined zero-dimensional proximities and zero-dimensional
compactifications.

Mrówka et al. [7] introduced the theory of δ-connectedness (or equiconnect-
edness) in proximity spaces. Consequently, Dimitrijević et al. [2, 3] defined
local δ-connectedness, δ-component and the treelike proximity spaces. In 1978,
Kohli [6] introduced the notion of sum connectedness in topological spaces.

We discuss sum δ-connectedness in proximity spaces in this paper. Some
necessary definitions and the results which are used in further sections, are re-
called in Section 2. In Section 3, sum δ-connectedness is defined and its relations
with other kinds of connectedness are determined. Several characterizations of

Received 16 December 2020 – Accepted 18 March 2021

http://dx.doi.org/10.4995/agt.2021.14809


B. Singh and D. Singh

it are established. It is shown that sum δ-connectedness is equivalent to local δ-
connectedness in a zero-dimensional proximity space. Further, the Stone-Čech
compactification of a separated proximity space X is sum δ-connected if and
only if X is sum δ-connected and it has finitely many δ-components. For a
sum δ-connected proximity space to be sum connected, a sufficient condition
is deduced. In the last section, weaker forms of sum δ-connectedness are de-
fined. Finally, if a sum δ-connected space is δ-padded, then it is also locally
δ-connected.

2. Preliminaries

Definition 2.1 ([9]). A binary relation δ on the power set P(X) of X is said
to be a proximity on X, if the following axioms are satisfied for all P , Q, R in
P(X):

(i) (φ, P) /∈ δ;
(ii) If P ∩ Q 6= φ, then (P, Q) ∈ δ;
(iii) If (P, Q) ∈ δ, then (Q, P) ∈ δ;
(iv) (P, Q ∪ R) ∈ δ if and only if (P, Q) ∈ δ or (P, R) ∈ δ;
(v) If (P, Q) /∈ δ, then there exists a subset R of X such that (P, R) /∈ δ and

(X\R, Q) /∈ δ.

The pair (X, δ) is called a proximity space.

Throughout this paper, we simply write proximity space (X, δ) as X when-
ever there is no confusion of the proximity δ.

Definition 2.2 ([8, 9]). A proximity space X is said to be separated if x = y
whenever ({x}, {y}) ∈ δ for x, y ∈ X.

Proposition 2.3 ([9]). Let X be a proximity space and P be a subset of X. If
P is δ-closed if and only if x ∈ P whenever ({x}, P) ∈ δ, then the collection of
the complements of all δ-closed sets forms a topology Tδ on X.

Proposition 2.4 ([9]). Let X be a proximity space. Then the closure C(P) of
P with respect to Tδ is given by C(P) = {x ∈ X : ({x}, P) ∈ δ}.

Corollary 2.5 ([9]). Let X be a proximity space. Then M ∈ Tδ if and only if
({x}, X\M) /∈ δ for every x ∈ M.

Using Proposition 2.4, a set F is δ-closed if C(F) = F . From Corollary 2.5,
a set U is δ-open, if ({x}, X\U) /∈ δ for every x ∈ U.

Definition 2.6 ([9]). Let X be a proximity space and T be a topology on X.
Then δ is said to be compatible with T if the generated topology Tδ and T are
equal, that is, Tδ = T .

Definition 2.7 ([9]). Let X be a proximity space. Then a subset N of X is
said to be a δ-neighbourhood of M ⊂ X if (M, X\N) /∈ δ. It is denoted by
M ≪δ N.

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Sum connectedness in proximity spaces

Definition 2.8 ([9]). Let (X, δ) and (Y, δ′) be two proximity spaces. Then
a map f : (X, δ) −→ (Y, δ′) is said to be δ-continuous ( or p-continuous ) if
(f(P), f(Q)) ∈ δ′ whenever (P, Q) ∈ δ, for all P, Q ⊂ X.

Definition 2.9 ([7]). Let X be a proximity space. Then X is said to be δ-
connected if every δ-continuous map from X to a discrete proximity space is
constant.

Theorem 2.10 ([7]). Let X be a proximity space. Then the following state-
ments are equivalent:

(i) X is δ-connected.
(ii) (P, X\P) ∈ δ for each nonempty subset P with P 6= X.
(iii) For every δ-continuous real-valued function f, the image f(X) is dense

in some interval of R.
(iv) If X = P ∪ Q and (P, Q) /∈ δ, then either P = φ or Q = φ.

Definition 2.11 ([2]). Let X be a proximity space and x ∈ X. Then the
δ-component of a point x is defined as the union of all δ-connected subsets of
X containing x. It is denoted by Cδ(x).

Definition 2.12 ([2]). Let X be a proximity space and x ∈ X. Then the
δ-quasi component of x is the equivalence class of x with respect to the equiv-
alence relation ∼ defined on X as “ x ∼ y if and only if there do not exist the
sets M, N such that x ∈ M and y ∈ N with X = M ∪ N and (M, N) /∈ δ”.

Definition 2.13 ([2]). A proximity space X is called locally δ-connected if for
every point x of X and for every δ-neighbourhood N of x, there exists some
δ-connected δ-neighbourhood M of x such that x ∈ M ⊂ N.

Definition 2.14 ([12]). Let (X, δ) be a proximity space and f : X −→ Y be
a surjective map, where Y is any set. Then the quotient proximity on Y is the
finest proximity such that the map f is δ-continuous. When Y has the quotient
proximity, f is called δ-quotient map.

Proposition 2.15 ([12]). Let (X, δ) be a proximity space and f : X −→ Y be
a surjective map, where Y be any set. Then the quotient proximity δ′ on Y is
given by P ≪δ′ Q if and only if for each binary rational s ∈ [0, 1], there is some
Ps ⊆ Y such that P0 = P, P1 = Q and s < t implies f

−1(Ps) ≪δ f
−1(Pt).

Proposition 2.16 ([12]). Let (X, δ) be a proximity space and f : X −→ Y be
a surjective map such that f−1(f(M)) = M for each δ-open set M of X, where
Y be any set. Then the quotient proximity δ′ on Y is given by (P, Q) ∈ δ′ if
and only if (f−1(P), f−1(Q)) ∈ δ.

Definition 2.17 ([1]). A proximity space X is said to be zero-dimensional if
the proximity δ satisfies the following axiom:

If (P, Q) /∈ δ, then there is a subset R of X such that (R, X\R) /∈ δ,
(P, R) /∈ δ and (X\R, Q) /∈ δ.

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B. Singh and D. Singh

Definition 2.18 ([6]). A topological space X is said to be sum connected at
x ∈ X, if there exists an open connected neighbourhood of x. If X is sum
connected at each of its points, then X is called sum connected.

Proposition 2.19 ([6]). Let X∗ be the Stone-Čech compactification of a Ty-
chonoff space X. Then X is sum connected and has finitely many components,
if X∗ is sum connected.

3. Sum δ-connectedness

Definition 3.1. A proximity space X is said to be sum δ-connected at x ∈ X
if there exists a δ-connected δ-open δ-neighbourhood of x. If X is sum δ-
connected at each of its points, then it is said to be sum δ-connected.

Definition 3.2. Let (Xi, δi)i∈I be a family of proximity spaces, where I is an
index set. A proximity space (X, δ) is said to be a far proximity sum of (Xi)i∈I
if X =

⋃
i∈I

Xi and (Xi, Xj) /∈ δ for all i 6= j in I with δ|Xi = δi for all i ∈ I.

Note that a proximity space X is sum δ-connected if and only if each of its
δ-component is δ-open. Therefore, every δ-connected proximity space is sum
δ-connected.

Example 3.3.

(i) Let X be any discrete proximity space with |X| ≥ 2. Then X is sum
δ-connected but not δ-connected.

(ii) Let X = (0, 1) ∪ (2, 3) with usual subspace proximity of R. Then X is
sum δ-connected but not δ-connected.

Every sum connected proximity space is sum δ-connected. But, converse
may not be true. However, in compact separated proximity spaces, the notion
of sum connectedness and sum δ-connectedness coincides.

Example 3.4. The space Q of rationals with the usual proximity is sum δ-
connected. But, it is not sum connected.

Every locally δ-connected proximity space is sum δ-connected. Converse
may not be true.

Example 3.5. Consider T = {(x, sin 1
x
) : 0 < x ≤ 1} ∪ {(0, y) : −1 ≤ y ≤ 1}

the closed Topologist’s Sine curve with subspace proximity induced from R2.
Let X be the far proximity sum of two copies of T . Then X is sum δ-connected
but it is neither δ-connected nor locally δ-connected.

Example 3.6. Let X = {0} ∪ { 1
n
: n ∈ N} be a proximity space. Since each

{ 1
n
} is δ-clopen in X, there does not exists any δ-connected δ-neighbourhood of

0 in X because every δ-neighbourhood of 0 contains infinitely many members
of X\{0}. Thus, X is not sum δ-connected at 0.

Thus, we have following relationship among several connectednesses in prox-
imity space.

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Sum connectedness in proximity spaces

connected =⇒ sum connected ⇐= locally connected
⇓ ⇓ ⇓

δ − connected =⇒ sum δ − connected ⇐= locally δ − connected

The following theorem gives some necessary and sufficient conditions for sum
δ-connectedness.

Theorem 3.7. For a proximity space X, the following statements are equiva-
lent:

(i) X is sum δ-connected.
(ii) For each x ∈ X and each δ-clopen set U which contains x, there exists a

δ-open δ-connected set W containing x such that W ⊂ U.
(iii) δ-components of δ-clopen sets in X are δ-open in X.

Proof. (i) =⇒ (ii). Let x ∈ X and U be a δ-clopen set such that x ∈ U. Let
Cδ(x) be the δ-component of X containing x. By hypothesis, Cδ(x) is δ-open.
So, Cδ(x) ∩ U is δ-clopen. Therefore, ((Cδ(x) ∩ U), Cδ(x)\(Cδ(x) ∩ U)) /∈ δ as
Cδ(x) ∩ U ⊂ Cδ(x) ⊂ X. Also, since Cδ(x) is δ-connected, Cδ(x) ∩ U = Cδ(x).
Hence, Cδ(x) is a δ-open δ-connected such that Cδ(x) ⊂ U.

(ii) =⇒ (iii). Let U be any δ-clopen set in X and Cδ be a δ-component of
U. Then, by hypothesis, for each x ∈ Cδ there exists a δ-open δ-connected set
W such that x ∈ W ⊂ U. Therefore, W ⊂ Cδ as Cδ is δ-component. Hence,
Cδ is δ-open.

(iii) =⇒ (i). Since X is δ-clopen, the result follows. �

Proposition 3.8. Let Y be a dense proximity subspace of X and x ∈ Y . Then
X is sum δ-connected at x if Y is sum δ-connected at x.

Proof. Let W be a δ-open δ-connected δ-neighbourhood of x in Y . Therefore,
W = U ∩ Y , where U is δ-open δ-neighbourhood of x in X. Thus, W ⊂ U and
U ⊂ ClX(U) = ClX(W) as Y is dense in X. Note that ClX(W) is δ-connected.
Hence, U is δ-open δ-connected δ-neighbourhood of x in X. �

Next example shows that the closure of sum δ-connected proximity space
may not be sum δ-connected.

Example 3.9. Let X = { 1
n
: n ∈ N} be a proximity subspace of R. Then each

δ-component { 1
n
} is δ-clopen in X. So, X is sum δ-connected. But, note that

Cl(X) = {0} ∪ { 1
n
: n ∈ N} is not sum δ-connected at 0 by Example 3.6.

Proposition 3.10. Let X be a sum δ-connected proximity space and f :
(X, δ) −→ (Y, δ∗) be a δ-quotient map such that f−1(f(U)) = U for each
δ-open subset U of X. Then Y is sum δ-connected.

Proof. Let Cδ be any δ-component of Y and y ∈ Cδ. We have to show that
(y, Y \Cδ) /∈ δ

∗. By definition of δ-quotient proximity δ∗, it suffices to show
that (f−1(y), X\f−1(Cδ)) /∈ δ. Let x ∈ f

−1(y), then the δ-component Cx of
x in X, be δ-open in X. Therefore, (z, X\Cx) /∈ δ for every z ∈ Cx. Since

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B. Singh and D. Singh

f is δ-continuous, f(Cx) is δ-connected. Thus, y = f(x) ∈ f(Cx) ∩ Cδ. So
f(Cx) ⊆ Cδ, which implies Cx ⊆ f

−1(Cδ). Then, (z, X\f
−1(Cδ)) /∈ δ for every

z ∈ Cx. In particular, (f
−1(y), X\f−1(Cδ) /∈ δ. �

Corollary 3.11. Let f : (X, δ) −→ (Y, δ∗) be a δ-continuous, δ-closed, sur-
jection such that f−1(f(U)) = U for each δ-open subset U of X. If X is sum
δ-connected, then Y is also sum δ-connected.

Proposition 3.12. Every δ-continuous, δ-open image of a sum δ-connected
proximity space is sum δ-connected.

Proof. Let f : (X, δ) −→ (Y, δ′) be a δ-continuous, δ-open, surjective map and
X be sum δ-connected. Let Cδ be a δ-component of Y and x ∈ f

−1(Cδ).
Then there is a δ-component Cx in X containing x which is δ-open. Since
f is δ-continuous and δ-open, f(Cx) ⊆ Cδ and f(Cx) is δ-open. Therefore,
(f(x), Y \f(Cx)) /∈ δ

′. Hence, (f(x), Y \Cδ) /∈ δ
′. �

Corollary 3.13. If the product of proximity spaces is sum δ-connected, then
each of its factor is also sum δ-connected.

The product of sum δ-connected proximity spaces need not be sum δ-connected
in general.

Example 3.14. Let X = {0, 1}ω be infinite product of two point discrete
proximity spaces. Then X is not discrete proximity space. Therefore, the δ-
component Cδ(x) of x in X is {x} itself, which is not δ-open. Hence, X is not
sum δ-connected.

Theorem 3.15. Let (X, δ) be a product of proximity spaces (Xi, δi)i∈I, where
I is an index set. Then X =

∏
i∈I

Xi is sum δ-connected if and only if each
Xi is sum δ-connected and all but finitely many Xi’s are δ-connected.

Proof. Let X be sum δ-connected. So, by Corollary 3.13, each Xi is sum δ-
connected. Now, suppose that all but finitely many Xi’s are not δ-connected.
Then any δ-component of X is not δ-open in X, which is a contradiction.

Conversely, assume that each Xi is sum δ-connected and all but finitely
many Xi’s are δ-connected. Let Cδ be any δ-component of X and pi be the
ith projection map. Then pi(Cδ) is δ-connected for each i ∈ I. Therefore,∏

i∈I
pi(Cδ) is also δ-connected. Thus, Cδ =

∏
i∈I

pi(Cδ). For each i ∈ I,
suppose Cδi be the δi-component of Xi containing pi(Cδ). Put C

′
δ =

∏
i∈I

Cδi.
If pi(Cδ) ( Cδi, then Cδ = C

′
δ as Cδ is δ-component of X. Thus, pi(Cδ) = Cδi

for each i ∈ I. Since all but finitely many Xi’s are δ-connected, pi(Cδ) = Cδi =
Xi for all but finitely many i ∈ I. Hence, Cδ is δ-open set in X. �

Theorem 3.16. Every far proximity sum of sum δ-connected proximity spaces
is sum δ-connected.

It can be easily shown that a δ-closed subspace of sum δ-connected proximity
space need not be sum δ-connected.

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Sum connectedness in proximity spaces

Corollary 3.17. A proximity space X is locally δ-connected if and only if every
δ-open subspace of X is sum δ-connected.

Theorem 3.18. Let X be a pseudocompact, separated, sum δ-connected prox-
imity space. Then it has at most finitely many δ-components.

Proof. Suppose X has infinitely many δ-components. Since collection of δ-
components of X is locally finite and each δ-component of X is δ-open, we
have a locally finite collection of non-empty δ-open sets which is not finite, a
contradiction. �

Corollary 3.19. If X is compact sum δ-connected proximity space, then it has
at most finitely many δ-components.

Corollary 3.20. If X is Lindelof (or separable) sum δ-connected proximity
space, then it has at most countably many δ-components.

Theorem 3.21. Every separated, zero-dimensional, sum δ-connected proximity
space is discrete.

Proof. Let X be any separated, zero-dimensional, sum δ-connected proximity
space. Let S be a subset of X such that x, y ∈ S with x 6= y. Therefore,
({x}, {y}) /∈ δ. Then, there exists C ⊂ X such that (C, X\C) /∈ δ, ({x}, C) /∈ δ
and (X\C, {y}) /∈ δ. So, (C, S\C) /∈ δ which implies S is not δ-connected.
Hence, every δ-component of X is singleton. As X is sum δ-connected, each
singleton of X is δ-open. �

Next theorem shows that in a zero-dimensional proximity space, local δ-
connectedness and sum δ-connectedness are equivalent.

Proposition 3.22. A zero-dimensional proximity space X is locally δ-connected
if and only if it is sum δ-connected.

Proof. Necessity is obvious. For the sufficient part, let X be sum δ-connected.
Let x ∈ X and U be a δ-neighbourhood of x. Therefore, there exists C ⊂ X
such that (C, X\C) /∈ δ, ({x}, X\C) /∈ δ and (C, X\U) /∈ δ. Thus, C is δ-
clopen and x ∈ C ⊂ U. So, by Theorem 3.7, there exists a δ-open δ-connected
set W such that x ∈ W ⊂ C ⊂ U. Hence, X is locally δ-connected. �

Now, we find the relation of sum δ-connectedness of proximity space with
its Stone-Čech compactification.

Theorem 3.23. Let (X∗, δ∗) be the Stone-Čech compactification of the sepa-
rated proximity space (X, δ). Then X∗ is sum δ-connected if and only if X is
sum δ-connected and has finitely many δ-components.

Proof. Let X∗ be sum δ-connected. Then, by Corollary 3.19, it has finitely
many δ-components. So, X∗ =

⋃n
i=1

Ciδ, where C
i
δ is a δ-component of X

∗ for
each 1 ≤ i ≤ n. Therefore, X =

⋃n
i=1

(Ciδ ∩ X). As each C
i
δ ∩ X is δ-open

in X and (Ciδ ∩ X, C
j
δ
∩ X) /∈ δ by using hypothesis, it suffices to show that

each Ciδ ∩ X is δ-connected. Let C
i
δ ∩ X = P ∪ Q with (P, Q) /∈ δ

∗. Note that

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B. Singh and D. Singh

Clδ∗(C
i
δ∩X) = C

i
δ because C

i
δ is δ-open in X

∗ and X is dense in X∗. Therefore,
Ciδ = Clδ∗(C

i
δ ∩ X) = Clδ∗(P) ∪ Clδ∗(Q) with (Clδ∗ (P), Clδ∗(Q)) /∈ δ

∗. Thus,
Ciδ is not δ-connected, a contradiction.

Conversely, assume X is sum δ-connected and has finitely many δ-components.
Therefore, X =

⋃n
i=1

Ciδ where C
i
δ is a δ-component of X for each 1 ≤ i ≤ n.

Thus, X∗ = Clδ∗(X) =
⋃n

i=1
Clδ∗(C

i
δ). Since, (C

i
δ, C

j
δ
) /∈ δ for i 6= j,

(Clδ∗(C
i
δ), Clδ∗ (C

j
δ
)) /∈ δ∗. Note that each Clδ∗(C

i
δ) is δ-connected in X

∗.
Thus, each Clδ∗(C

i
δ) is a δ-component in X

∗. Since δ-components in X∗ are
finite, hence X∗ is sum δ-connected. �

Corollary 3.24. If X is pseudocompact, separated and sum δ-connected prox-
imity space, then it’s Stone-Čech compactification X∗ is also sum δ-connected.

Every sum connected proximity space is sum δ-connected. Following theo-
rem gives the sufficient condition for a sum δ-connected proximity space to be
sum connected.

Theorem 3.25. Let (X, T ) be a Tychonoff space. If X is sum δ-connected
and has finitely many δ-components with respect to any proximity δ compatible
with T , then X is sum connected. Moreover, it has at most finitely many
components.

Proof. Let S be the collection of all proximities which are compatible with T .
Let δ0 = sup S, then δ0 is also compatible with T . Therefore, by hypothesis,
X is sum δ-connected and has finitely many δ-components with respect to δ0.
Since δ0 = sup S, the compactification (X

∗, δ∗) corresponding to δ0 is Stone-
Čech compactification. So, by Theorem 3.23, X∗ is sum δ-connected. Thus,
X∗ is sum connected. By Proposition 2.19, X is sum connected and has finitely
many components. �

4. Weaker forms of sum δ-connectedness

In this section we give proximity versions of notions defined and considered
in [6].

Definition 4.1. Let X be a proximity space which contains a point x. Then
X is called :

(i) weakly sum δ-connected at x if there exists a δ-connected δ-neighbourhood
of x.

(ii) quasi sum δ-connected at x if the δ-quasi component which contains x
is a δ-neighbourhood of x.

(iii) δ-padded at x if for every δ-neighbourhood W of x there exist δ-open
sets U and V such that x ∈ U ⊆ Clδ(U) ⊆ V ⊆ W and V \Clδ(U) has
at most finitely many δ-components.

If a proximity space X is weakly sum δ-connected (or quasi sum δ-connected)
at each of its points, then the space X is called weakly sum δ-connected (or
quasi sum δ-connected). For a proximity space X,

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Sum connectedness in proximity spaces

sum δ-connected ⇒ weakly sum δ-connected ⇒ quasi sum δ-connected

Example 4.2. In R2, let Bn be the infinite broom containing all the closed line
segments joining the point ( 1

n
, 0) to the points {( 1

n+1
, 1
m
) : m = n, n + 1, · · · },

where n = 1, 2, · · · . Let B =
⋃

∞

n=1
Bn and A = {(x, 0) : 0 ≤ x ≤ 2} ∪ {(y,

1
n
) :

1 ≤ y ≤ 2 and n = 1, 2, · · ·}. Let X = A ∪ B. Then note that X is compact.
Therefore, connectedness is equivalent to δ-connectedness. Hence, X is weak
sum δ-connected but not sum δ-connected at (0, 0).

Lemma 4.3. Every δ-open δ-quasi component is a δ-component.

Proof. Let U be a δ-open δ-quasi component of proximity space X and x ∈ U.
Let V be the δ-component of x. Then V ⊂ U. Let y ∈ U\V . So, x ∼ y. Since
V is δ-closed in X and V ⊂ U, V is δ-closed in U. So, U\V is δ-open in U. As
U is δ-open in X, U\V is δ-open in X. Therefore, (U\V, X\(U\V )) /∈ δ. Thus,
X = (U\V ) ∪ (X\(U\V )) with (U\V, X\(U\V )) /∈ δ. Hence, x ≁ y which is a
contradiction. �

Proposition 4.4. For a given proximity space X, the following statements are
comparable:

(i) X is quasi sum δ-connected.
(ii) X is weakly sum δ-connected.
(iii) X is sum δ-connected.
(iv) δ-components of X are δ-open.
(v) δ-quasi components of X are δ-open.

Proof. By Lemma 4.3, δ-open δ-quasi component is a δ-component. There-
fore the statements (iv) and (v) are equivalent. The equivalence of (iv) with
(i), (ii), (iii) follows from the fact that a set is δ-open if and only if it is a
δ-neighbourhood of each of its points. �

Corollary 4.5. A proximity space X is sum δ-connected if and only if it is the
far proximity sum of its δ-components (δ-quasi components).

Corollary 4.6. Let X be a sum δ-connected proximity space. Then the map
f on X is δ-continuous if and only if it is δ-continuous on each of its δ-
component.

Corollary 4.7. Every locally δ-connected proximity space is the far proximity
sum of its δ-components (δ-quasi components).

Corollary 4.8. If X is sum δ-connected proximity space and U ⊂ X, then U
is a δ-component if and only if it is δ-quasi component. In particular, If Y
is a locally δ-connected proximity space and X ⊂ Y is δ-open, then U ⊂ X is
δ-component if and only if it is δ-quasi component.

Proof. By Proposition 4.4 (iv), δ-components and δ-quasi components coincide
in sum δ-connected proximity space. The last statement of corollary from
the fact that every locally δ-connected proximity space is sum δ-connected;

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B. Singh and D. Singh

and every δ-open subset of a locally δ-connected proximity space is locally
δ-connected. �

As in Example 3.5, sum δ-connected proximity space may not be locally δ-
conneted. But, if sum δ-connected proximity space is δ-padded, then it is also
locally δ-connected.

Proposition 4.9. Let X be a sum δ-connected proximity space and x ∈ X. If
X is δ-padded at x, then it is locally δ-connected at x.

Proof. Let N be a δ-open δ-neighbourhood of x. As X is sum δ-connected,
suppose that N is contained in δ-component Cδ. Since X is δ-padded at x,
there are δ-open δ-neighbourhoods W and V of x such that Clδ(W) ⊆ V ⊆ N
with V \Clδ(W) has only finitely many δ-components C

1
δ , C

2
δ , · · · , C

n
δ . Now for

each i, 1 ≤ i ≤ n, there exist a δ-quasi component Qiδ such that C
i
δ ⊆ Q

i
δ. We

show that each v ∈ V is in some Qiδ. If there is some v ∈ V such that v /∈ Q
i
δ for

each 1 ≤ i ≤ n, then for each i we have V = (V \Qiδ)∪Q
i
δ with (V \Q

i
δ, Q

i
δ) /∈ δ.

Let Wi = V \Q
i
δ for each 1 ≤ i ≤ n and M =

⋂
i
Wi. Since (V \Q

i
δ, Q

i
δ) /∈ δ

for each 1 ≤ i ≤ n, (M, Qiδ) /∈ δ. Note that Cδ\M =
⋃

i
Cδ\Wi and for

each i, Cδ\Wi = (Cδ\V ) ∪ Q
i
δ. As V is δ-open in Cδ, (V, Cδ\V ) /∈ δ which

implies (M, Cδ\V ) /∈ δ. Thus, (M, (Cδ\V ) ∪ Q
i
δ) /∈ δ, that is, (M, Cδ\Wi) /∈ δ

for each i. Therefore, (M, Cδ\M) /∈ δ. Therefore, Cδ is not δ-connected, a
contradiction. Thus, each v ∈ V is in some Qiδ. Therefore, V has only finitely
many δ-quasi components and each of them is δ-open. Thus, each δ-quasi
component is a δ-component. Hence, δ-component of x in V is δ-connected
δ-open neighbourhood of x contained in N. �

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