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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 931656, 17 pages
http://dx.doi.org/10.1155/2012/931656
Research Article

On Upper and Lower 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -Continuous Multifunctions

Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand

Received 9 May 2012; Accepted 24 June 2012

Academic Editor: B. N.Β Mandal

Copyright Β© 2012 Chawalit Boonpok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new class of multifunctions, called upper (lower) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous multifunctions, has been defined and studied. Some characterizations and several properties concerning upper (lower) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous multifunctions are obtained. The relationships between upper (lower) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous multifunctions and some known concepts are also discussed.

1. Introduction

General topology has shown its fruitfulness in both the pure and applied directions. In reality it is used in data mining, computational topology for geometric design and molecular design, computer-aided design, computer-aided geometric design, digital topology, information system, and noncommutative geometry and its application to particle physics. One can observe the influence made in these realms of applied research by general topological spaces, properties, and structures. Continuity is a basic concept for the study of general topological spaces. This concept has been extended to the setting of multifunctions and has been generalized by weaker forms of open sets such as 𝛼 -open sets [1], semiopen sets [2], preopen sets [3], 𝛽 -open sets [4], and semi-preopen sets [5]. Multifunctions and of course continuous multifunctions stand among the most important and most researched points in the whole of the mathematical science. Many different forms of continuous multifunctions have been introduced over the years. Some of them are semicontinuity [6], 𝛼 -continuity [7], precontinuity [8], quasicontinuity [9], 𝛾 -continuity [10], and 𝛿 -precontinuity [11]. Most of these weaker forms of continuity, in ordinary topology such as 𝛼 -continuity and 𝛽 -continuity, have been extended to multifunctions [1215]. Császár [16] introduced the notions of generalized topological spaces and generalized neighborhood systems. The classes of topological spaces and neighborhood systems are contained in these classes, respectively. Specifically, he introduced the notions of continuous functions on generalized topological spaces and investigated the characterizations of generalized continuous functions. Kanibir and Reilly [17] extended these concepts to multifunctions. The purpose of the present paper is to define upper (lower) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous multifunctions and to obtain several characterizations of upper (lower) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous multifunctions and several properties of such multifunctions. Moreover, the relationships between upper (lower) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous multifunctions and some known concepts are also discussed.

2. Preliminaries

Let 𝑋 be a nonempty set, and denote 𝒫 ( 𝑋 ) the power set of 𝑋 . We call a class πœ‡ βŠ† 𝒫 ( 𝑋 ) a generalized topology (briefly, GT) on 𝑋 if βˆ… ∈ πœ‡ , and an arbitrary union of elements of πœ‡ belongs to πœ‡ [16]. A set 𝑋 with a GT πœ‡ on it is said to be a generalized topological space (briefly, GTS) and is denoted by ( 𝑋 , πœ‡ ) . For a GTS ( 𝑋 , πœ‡ ) , the elements of πœ‡ are called πœ‡ -open sets and the complements of πœ‡ -open sets are called πœ‡ -closed sets. For π’œ βŠ† 𝑋 , we denote by 𝑐 πœ‡ ( π’œ ) the intersection of all πœ‡ -closed sets containing π’œ and by 𝑖 πœ‡ ( π’œ ) the union of all πœ‡ -open sets contained in π’œ . Then, we have 𝑖 πœ‡ ( 𝑖 πœ‡ ( π’œ ) ) = 𝑖 πœ‡ ( π’œ ) , 𝑐 πœ‡ ( 𝑐 πœ‡ ( π’œ ) ) = 𝑐 πœ‡ ( π’œ ) , and 𝑖 πœ‡ ( π’œ ) = 𝑋 βˆ’ 𝑐 πœ‡ ( 𝑋 βˆ’ π’œ ) . According to [18], for π’œ βŠ† 𝑋 and π‘₯ ∈ 𝑋 , we have π‘₯ ∈ 𝑐 πœ‡ ( π’œ ) if and only if π‘₯ ∈ 𝑀 ∈ πœ‡ implies 𝑀 ∩ π’œ β‰  βˆ… . Let ℬ βŠ† 𝒫 ( 𝑋 ) satisfy βˆ… ∈ ℬ . Then all unions of some elements of ℬ constitute a GT πœ‡ ( ℬ ) , and ℬ is said to be a base for πœ‡ ( ℬ ) [19]. Let πœ‡ be a GT on a set 𝑋 β‰  βˆ… . Observe that 𝑋 ∈ πœ‡ must not hold; if all the same 𝑋 ∈ πœ‡ , then we say that the GT πœ‡ is strong [20]. In general, let β„³ πœ‡ denote the union of all elements of πœ‡ ; of course, β„³ πœ‡ ∈ πœ‡ and β„³ πœ‡ = 𝑋 if and only if πœ‡ is a strong GT. Let us now consider those GT’s πœ‡ that satisfy the folllowing condition: if 𝑀 , 𝑀 ξ…ž ∈ πœ‡ , then 𝑀 ∩ 𝑀 ξ…ž ∈ πœ‡ . We will call such a GT quasitopology (briefly QT) [21]; the QTs clearly are very near to the topologies.

A subset β„› of a generalized topological space ( 𝑋 , πœ‡ ) is said to be πœ‡ π‘Ÿ -open [18] (resp. πœ‡ π‘Ÿ -closed) if β„› = 𝑖 πœ‡ ( 𝑐 πœ‡ ( β„› ) ) (resp. β„› = 𝑐 πœ‡ ( 𝑖 πœ‡ ( β„› ) ) ). A subset π’œ of a generalized topological space ( 𝑋 , πœ‡ ) is said to be πœ‡ -semiopen [22] (resp. πœ‡ -preopen, πœ‡ -α-open,  and πœ‡ -β-open) if π’œ βŠ† 𝑐 πœ‡ ( 𝑖 πœ‡ ( π’œ ) ) (resp. π’œ βŠ† 𝑖 πœ‡ ( 𝑐 πœ‡ ( π’œ ) ) , π’œ βŠ† 𝑖 πœ‡ ( 𝑐 πœ‡ ( 𝑖 πœ‡ ( π’œ ) ) ) , π’œ βŠ† 𝑐 πœ‡ ( 𝑖 πœ‡ ( 𝑐 πœ‡ ( π’œ ) ) ) ). The family of all πœ‡ -semiopen (resp. πœ‡ -preopen, πœ‡ - 𝛼 -open, πœ‡ - 𝛽 -open) sets of 𝑋 containing a point π‘₯ ∈ 𝑋 is denoted by 𝜎 ( πœ‡ , π‘₯ ) (resp. πœ‹ ( πœ‡ , π‘₯ ) , 𝛼 ( πœ‡ , π‘₯ ) , and 𝛽 ( πœ‡ , π‘₯ ) ). The family of all πœ‡ -semiopen (resp. πœ‡ -preopen, πœ‡ - 𝛼 -open, πœ‡ - 𝛽 -open) sets of 𝑋 is denoted by 𝜎 ( πœ‡ ) (resp. πœ‹ ( πœ‡ ) , 𝛼 ( πœ‡ ) , and 𝛽 ( πœ‡ ) ). It is shown in [22, Lemma  2.1] that 𝛼 ( πœ‡ ) = 𝜎 ( πœ‡ ) ∩ πœ‹ ( πœ‡ ) and it is obvious that 𝜎 ( πœ‡ ) βˆͺ πœ‹ ( πœ‡ ) βŠ† 𝛽 ( πœ‡ ) . The complement of a πœ‡ -semiopen (resp. πœ‡ -preopen, πœ‡ - 𝛼 -open, and πœ‡ - 𝛽 -open) set is said to be πœ‡ -semiclosed (resp. πœ‡ -preclosed, πœ‡ -α-closed, and πœ‡ -β-closed).

The intersection of all πœ‡ -semiclosed (resp. πœ‡ -preclosed, πœ‡ - 𝛼 -closed, and   πœ‡ - 𝛽 -closed) sets of 𝑋 containing π’œ is denoted by 𝑐 𝜎 ( π’œ ) . 𝑐 πœ‹ ( π’œ ) , 𝑐 𝛼 ( π’œ ) , and 𝑐 𝛽 ( π’œ ) are defined similarly. The union of all πœ‡ - 𝛽 -open sets of 𝑋 contained in π’œ is denoted by 𝑖 𝛽 ( π’œ ) .

Now let 𝐾 β‰  βˆ… be an index set, 𝑋 π‘˜ β‰  βˆ… for π‘˜ ∈ 𝐾 , and ∏ 𝑋 = π‘˜ ∈ 𝐾 𝑋 π‘˜ the Cartesian product of the sets 𝑋 π‘˜ . We denote by 𝑝 π‘˜ the projection 𝑝 π‘˜ ∢ 𝑋 β†’ 𝑋 π‘˜ . Suppose that, for π‘˜ ∈ 𝐾 , 𝑒 π‘˜ is a given GT on 𝑋 π‘˜ . Let us consider all sets of the form ∏ π‘˜ ∈ 𝐾 𝑋 π‘˜ , where 𝑀 π‘˜ ∈ πœ‡ π‘˜ and, with the exception of a finite number of indices π‘˜ , 𝑀 π‘˜ = 𝑍 π‘˜ = 𝑀 πœ‡ π‘˜ . We denote by ℬ the collection of all these sets. Clearly βˆ… ∈ ℬ so that we can define a GT πœ‡ = πœ‡ ( ℬ ) having ℬ for base. We call πœ‡ the product [23] of the GT’s πœ‡ π‘˜ and denote it by 𝐏 π‘˜ ∈ 𝐾 πœ‡ π‘˜ .

Let us write 𝑖 = 𝑖 πœ‡ , 𝑐 = 𝑐 πœ‡ , 𝑖 π‘˜ = 𝑖 πœ‡ π‘˜ , and 𝑐 π‘˜ = 𝑐 πœ‡ π‘˜ . Consider in the following 𝐴 π‘˜ βŠ† 𝑋 π‘˜ , ∏ 𝐴 = π‘˜ ∈ 𝐾 𝐴 π‘˜ , ∏ π‘₯ ∈ π‘˜ ∈ 𝐾 𝑋 π‘˜ , and π‘₯ π‘˜ = 𝑝 π‘˜ ( π‘₯ ) .

Proposition 2.1 (see [23]). One has ∏ 𝑐 𝐴 = π‘˜ ∈ 𝐾 𝑐 π‘˜ 𝐴 π‘˜ .

Proposition 2.2 (see [24]). Let ∏ 𝐴 = π‘˜ ∈ 𝐾 𝐴 π‘˜ βŠ† ∏ π‘˜ ∈ 𝐾 𝑋 π‘˜ , and let 𝐾 0 be a finite subset of 𝐾 . If 𝐴 π‘˜ ∈ { 𝑀 π‘˜ , 𝑋 π‘˜ } for each π‘˜ ∈ 𝐾 βˆ’ 𝐾 0 , then ∏ 𝑖 𝐴 = π‘˜ ∈ 𝐾 𝑖 π‘˜ 𝐴 π‘˜ .

Proposition 2.3 (see [23]). The projection 𝑝 π‘˜ is ( πœ‡ , πœ‡ π‘˜ ) -open.

Proposition 2.4 (see [23]). If every πœ‡ π‘˜ is strong, then πœ‡ is strong and 𝑝 π‘˜ is ( πœ‡ , πœ‡ π‘˜ ) -continuous for π‘˜ ∈ 𝐾 .

Throughout this paper, the spaces ( 𝑋 , πœ‡ 𝑋 ) and ( π‘Œ , πœ‡ π‘Œ ) (or simply 𝑋 and π‘Œ ) always mean generalized topological spaces. By a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , we mean a point-to-set correspondence from 𝑋 into π‘Œ , and we always assume that 𝐹 ( π‘₯ ) β‰  βˆ… for all π‘₯ ∈ 𝑋 . For a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , we will denote the upper and lower inverse of a set 𝐺 of π‘Œ by 𝐹 + ( 𝐺 ) and 𝐹 βˆ’ ( 𝐺 ) , respectively, that is 𝐹 + ( 𝐺 ) = { π‘₯ ∈ 𝑋 ∢ 𝐹 ( π‘₯ ) βŠ† 𝐺 } and 𝐹 βˆ’ ( 𝐺 ) = { π‘₯ ∈ 𝑋 ∢ 𝐹 ( π‘₯ ) ∩ 𝐺 β‰  βˆ… } . In particular, 𝐹 βˆ’ ( 𝑦 ) = { π‘₯ ∈ 𝑋 ∢ 𝑦 ∈ 𝐹 ( π‘₯ ) } for each point 𝑦 ∈ π‘Œ . For each 𝐴 βŠ† 𝑋 , 𝐹 ( 𝐴 ) = βˆͺ π‘₯ ∈ 𝐴 𝐹 ( π‘₯ ) . Then, 𝐹 is said to be a surjection if 𝐹 ( 𝑋 ) = π‘Œ , or equivalently, if for each 𝑦 ∈ π‘Œ there exists an π‘₯ ∈ 𝑋 such that 𝑦 ∈ 𝐹 ( π‘₯ ) .

3. Upper and Lower 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -Continuous Multifunctions

Definition 3.1. Let ( 𝑋 , πœ‡ 𝑋 ) and ( π‘Œ , πœ‡ π‘Œ ) be generalized topological spaces. A multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ is said to be(1)upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous at a point π‘₯ ∈ 𝑋 if, for each πœ‡ π‘Œ -open set 𝑉 of π‘Œ containing 𝐹 ( π‘₯ ) , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ ) βŠ† 𝑉 ,(2)lower 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous at a point π‘₯ ∈ 𝑋 if, for each πœ‡ π‘Œ -open set 𝑉 of π‘Œ such that 𝐹 ( π‘₯ ) ∩ 𝑉 β‰  βˆ… , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( 𝑧 ) ∩ 𝑉 β‰  βˆ… for every 𝑧 ∈ π‘ˆ ,(3)upper (resp. lower) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous if 𝐹 has this property at each point of 𝑋 .

Lemma 3.2. Let 𝐴 be a subset of a generalized topological space ( 𝑋 , πœ‡ 𝑋 ) . Then,(1) π‘₯ ∈ 𝑐 𝛽 𝑋 ( 𝐴 ) if and only if 𝐴 ∩ π‘ˆ β‰  βˆ… for each π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) ,(2) 𝑐 𝛽 𝑋 ( 𝑋 βˆ’ 𝐴 ) = 𝑋 βˆ’ 𝑖 𝛽 𝑋 ( 𝐴 ) ,(3) 𝐴 is πœ‡ 𝑋 - 𝛽 -closed in 𝑋 if and only if 𝐴 = 𝑐 𝛽 𝑋 ( 𝐴 ) ,(4) 𝑐 𝛽 𝑋 ( 𝐴 ) is πœ‡ 𝑋 - 𝛽 -closed in 𝑋 .

Theorem 3.3. For a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , the following properties are equivalent:(1) 𝐹 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(2) 𝐹 + ( 𝑉 ) = 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) for every πœ‡ π‘Œ - 𝛽 -open set 𝑉 of π‘Œ ,(3) 𝐹 βˆ’ ( 𝑀 ) = 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑀 ) ) for every πœ‡ π‘Œ - 𝛽 -closed set 𝑀 of π‘Œ ,(4) 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝐴 ) ) βŠ† 𝐹 βˆ’ ( 𝑐 𝛽 π‘Œ ( 𝐴 ) ) for every subset 𝐴 of π‘Œ ,(5) 𝐹 + ( 𝑖 𝛽 π‘Œ ( 𝐴 ) ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝐴 ) ) for every subset 𝐴 of π‘Œ .

Proof. ( 1 ) β‡’ ( 2 ) Let 𝑉 be any πœ‡ π‘Œ - 𝛽 -open set of π‘Œ and π‘₯ ∈ 𝐹 + ( 𝑉 ) . Then 𝐹 ( π‘₯ ) βŠ† 𝑉 . There exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 ) containing π‘₯ such that 𝐹 ( π‘ˆ ) βŠ† 𝑉 . Thus π‘₯ ∈ π‘ˆ βŠ† 𝐹 + ( 𝑉 ) . This implies that π‘₯ ∈ 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) . This shows that 𝐹 + ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) . We have 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) βŠ† 𝐹 + ( 𝑉 ) . Therefore, 𝐹 + ( 𝑉 ) = 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) .
( 2 ) β‡’ ( 3 ) Let 𝑀 be any πœ‡ π‘Œ - 𝛽 -closed set of π‘Œ . Then, π‘Œ βˆ’ 𝑀 is πœ‡ π‘Œ - 𝛽 -open set, and we have 𝑋 βˆ’ 𝐹 βˆ’ ( 𝑀 ) = 𝐹 + ( π‘Œ βˆ’ 𝑀 ) = 𝑖 𝛽 𝑋 ( 𝐹 + ( π‘Œ βˆ’ 𝑀 ) ) = 𝑖 𝛽 𝑋 ( 𝑋 βˆ’ 𝐹 βˆ’ ( 𝑀 ) ) = 𝑋 βˆ’ 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑀 ) ) . Therefore, we obtain 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑀 ) ) = 𝐹 βˆ’ ( 𝑀 ) .
( 3 ) β‡’ ( 4 ) Let 𝐴 be any subset of π‘Œ . Since 𝑐 𝛽 π‘Œ ( 𝐴 ) is πœ‡ π‘Œ - 𝛽 -closed, we obtain 𝐹 βˆ’ ( 𝐴 ) βŠ† 𝐹 βˆ’ ( 𝑐 𝛽 π‘Œ ( 𝐴 ) ) = 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑐 𝛽 π‘Œ ( 𝐴 ) ) ) and 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝐴 ) ) βŠ† 𝐹 βˆ’ ( 𝑐 𝛽 π‘Œ ( 𝐴 ) ) .
( 4 ) β‡’ ( 5 ) Let 𝐴 be any subset of π‘Œ . We have 𝑋 βˆ’ 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝐴 ) ) = 𝑐 𝛽 𝑋 ( 𝑋 βˆ’ 𝐹 + ( 𝐴 ) ) = 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( π‘Œ βˆ’ 𝐴 ) ) βŠ† 𝐹 βˆ’ ( 𝑐 𝛽 π‘Œ ( π‘Œ βˆ’ 𝐴 ) ) = 𝐹 βˆ’ ( π‘Œ βˆ’ 𝑖 𝛽 π‘Œ ( 𝐴 ) ) = 𝑋 βˆ’ 𝐹 + ( 𝑖 𝛽 π‘Œ ( 𝐴 ) ) . Therefore, we obtain 𝐹 + ( 𝑖 𝛽 π‘Œ ( 𝐴 ) ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝐴 ) ) .
( 5 ) β‡’ ( 1 ) Let π‘₯ ∈ 𝑋 and 𝑉 be any πœ‡ π‘Œ - 𝛽 -open set of π‘Œ containing 𝐹 ( π‘₯ ) . Then π‘₯ ∈ 𝐹 + ( 𝑉 ) = 𝐹 + ( 𝑖 𝛽 π‘Œ ( 𝑉 ) ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) . There exists a πœ‡ 𝑋 - 𝛽 -open set π‘ˆ of 𝑋 containing π‘₯ such that π‘ˆ βŠ† 𝐹 + ( 𝑉 ) ; hence 𝐹 ( π‘ˆ ) βŠ† 𝑉 . This implies that 𝐹 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.

Theorem 3.4. For a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , the following properties are equivalent:(1) 𝐹 is lower 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(2) 𝐹 βˆ’ ( 𝑉 ) = 𝑖 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑉 ) ) for every πœ‡ π‘Œ - 𝛽 -open set 𝑉 of π‘Œ ,(3) 𝐹 + ( 𝑀 ) = 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝑀 ) ) for every πœ‡ π‘Œ - 𝛽 -closed set 𝑀 of π‘Œ ,(4) 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝐴 ) ) βŠ† 𝐹 + ( 𝑐 𝛽 π‘Œ ( 𝐴 ) ) for every subset 𝐴 of π‘Œ ,(5) 𝐹 ( 𝑐 𝛽 𝑋 ( 𝐴 ) ) βŠ† 𝑐 𝛽 π‘Œ ( 𝐹 ( 𝐴 ) ) for every subset 𝐴 of 𝑋 ,(6) 𝐹 βˆ’ ( 𝑖 𝛽 π‘Œ ( 𝐴 ) ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝐴 ) ) for every subset 𝐴 of π‘Œ .

Proof. We prove only the implications ( 4 ) β‡’ ( 5 ) and ( 5 ) β‡’ ( 6 ) with the proofs of the other being similar to those of Theorem 3.3.
( 4 ) β‡’ ( 5 ) Let 𝐴 be any subset of 𝑋 . By (4), we have 𝑐 𝛽 𝑋 ( 𝐴 ) βŠ† 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝐹 ( 𝐴 ) ) ) βŠ† 𝐹 + ( 𝑐 𝛽 π‘Œ ( 𝐹 ( 𝐴 ) ) ) and 𝐹 ( 𝑐 𝛽 𝑋 ( 𝐴 ) ) βŠ† 𝑐 𝛽 π‘Œ ( 𝐹 ( 𝐴 ) ) .
( 5 ) β‡’ ( 6 ) Let 𝐴 be any subset of π‘Œ . By (5), we have 𝐹 ( 𝑐 𝛽 𝑋 ( 𝐹 + ( π‘Œ βˆ’ 𝐴 ) ) ) βŠ† 𝑐 𝛽 π‘Œ ( 𝐹 ( 𝐹 + ( π‘Œ βˆ’ 𝐴 ) ) ) βŠ† 𝑐 𝛽 π‘Œ ( π‘Œ βˆ’ 𝐴 ) = π‘Œ βˆ’ 𝑖 𝛽 π‘Œ ( 𝐴 ) and 𝐹 ( 𝑐 𝛽 𝑋 ( 𝐹 + ( π‘Œ βˆ’ 𝐴 ) ) ) = 𝐹 ( 𝑐 𝛽 𝑋 ( 𝑋 βˆ’ 𝐹 βˆ’ ( 𝐴 ) ) ) = 𝐹 ( 𝑋 βˆ’ 𝑖 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝐴 ) ) ) . This implies that 𝐹 βˆ’ ( 𝑖 𝛽 π‘Œ ( 𝐴 ) ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝐴 ) ) .

Definition 3.5. A generalized topological space ( 𝑋 , πœ‡ 𝑋 ) is said to be πœ‡ 𝑋 - 𝛽 -compact if every cover of 𝑋 by πœ‡ 𝑋 - 𝛽 -open sets has a finite subcover.

A subset 𝑀 of a generalized topological space ( 𝑋 , πœ‡ 𝑋 ) is said to be πœ‡ 𝑋 - 𝛽 -compact if every cover of 𝑀 by πœ‡ 𝑋 - 𝛽 -open sets has a finite subcover.

Theorem 3.6. Let ( 𝑋 , πœ‡ 𝑋 ) be a generalized topological space and ( π‘Œ , πœ‡ π‘Œ ) a quasitopological space. If 𝐹 ∢ 𝑋 β†’ π‘Œ is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous multifunction such that 𝐹 ( π‘₯ ) is πœ‡ π‘Œ - 𝛽 -compact for each π‘₯ ∈ 𝑋 and 𝑀 is a πœ‡ 𝑋 - 𝛽 -compact set of 𝑋 , then 𝐹 ( 𝑀 ) is πœ‡ π‘Œ - 𝛽 -compact.

Proof. Let { 𝑉 𝛾 ∢ 𝛾 ∈ Ξ“ } be any cover of 𝐹 ( 𝑀 ) by πœ‡ π‘Œ - 𝛽 -open sets. For each π‘₯ ∈ 𝑀 , 𝐹 ( π‘₯ ) is πœ‡ π‘Œ - 𝛽 -compact and there exists a finite subset Ξ“ ( π‘₯ ) of Ξ“ such that 𝐹 ( π‘₯ ) βŠ† βˆͺ { 𝑉 𝛾 ∢ 𝛾 ∈ Ξ“ ( π‘₯ ) } . Now, set 𝑉 ( π‘₯ ) = βˆͺ { 𝑉 𝛾 ∢ 𝛾 ∈ Ξ“ ( π‘₯ ) } . Then we have 𝐹 ( π‘₯ ) βŠ† 𝑉 ( π‘₯ ) and 𝑉 ( π‘₯ ) is πœ‡ π‘Œ - 𝛽 -open set of π‘Œ . Since 𝐹 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous, there exists a πœ‡ 𝑋 - 𝛽 -open set π‘ˆ ( π‘₯ ) containing π‘₯ such that 𝐹 ( π‘ˆ ( π‘₯ ) ) βŠ† 𝑉 ( π‘₯ ) . The family { π‘ˆ ( π‘₯ ) ∢ π‘₯ ∈ 𝑀 } is a cover of 𝑀 by πœ‡ 𝑋 - 𝛽 -open sets. Since 𝑀 is πœ‡ 𝑋 - 𝛽 -compact, there exists a finite number of points, say, π‘₯ 1 , π‘₯ 2 , … , π‘₯ 𝑛 in 𝑀 such that 𝑀 βŠ† βˆͺ { π‘ˆ ( π‘₯ π‘š ) ∢ π‘₯ π‘š ∈ 𝑀 , 1 ≀ π‘š ≀ 𝑛 } . Therefore, we obtain 𝐹 ( 𝑀 ) βŠ† βˆͺ { 𝐹 ( π‘ˆ ( π‘₯ π‘š ) ) ∢ π‘₯ π‘š ∈ 𝑀 , 1 ≀ π‘š ≀ 𝑛 } βŠ† βˆͺ { 𝑉 𝛾 ∢ 𝛾 ∈ 𝛾 ( π‘₯ π‘š ) , π‘₯ π‘š ∈ 𝑀 , 1 ≀ π‘š ≀ 𝑛 } . This shows that 𝐹 ( 𝑀 ) is πœ‡ π‘Œ - 𝛽 -compact.

Corollary 3.7. Let ( 𝑋 , πœ‡ 𝑋 ) be a generalized topological space and ( π‘Œ , πœ‡ π‘Œ ) a quasitopological space. If 𝐹 ∢ 𝑋 β†’ π‘Œ is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous surjective multifunction such that 𝐹 ( π‘₯ ) is πœ‡ π‘Œ - 𝛽 -compact for each π‘₯ ∈ 𝑋 and ( 𝑋 , πœ‡ 𝑋 ) is πœ‡ 𝑋 - 𝛽 -compact, then ( π‘Œ , πœ‡ π‘Œ ) is πœ‡ π‘Œ - 𝛽 -compact.

Definition 3.8. A subset 𝐴 of a generalized topological space ( 𝑋 , πœ‡ 𝑋 ) is said to be πœ‡ 𝑋 - 𝛽 -clopen if 𝐴 is πœ‡ 𝑋 - 𝛽 -closed and πœ‡ 𝑋 - 𝛽 -open.

Definition 3.9. A generalized topological space ( 𝑋 , πœ‡ 𝑋 ) is said to be πœ‡ 𝑋 - 𝛽 -connected if 𝑋 can not be written as the union of two nonempty disjoint πœ‡ 𝑋 - 𝛽 -open sets.

Theorem 3.10. Let 𝐹 ∢ 𝑋 β†’ π‘Œ be upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous surjective multifunction. If ( 𝑋 , πœ‡ 𝑋 ) is πœ‡ 𝑋 - 𝛽 -connected and 𝐹 ( π‘₯ ) is πœ‡ π‘Œ - 𝛽 -connected for each π‘₯ ∈ 𝑋 , then ( π‘Œ , πœ‡ π‘Œ ) is πœ‡ π‘Œ - 𝛽 -connected.

Proof. Suppose that ( π‘Œ , πœ‡ π‘Œ ) is not πœ‡ π‘Œ - 𝛽 -connected. There exist nonempty πœ‡ π‘Œ - 𝛽 -open sets π‘ˆ and 𝑉 of π‘Œ such that π‘ˆ βˆͺ 𝑉 = π‘Œ and π‘ˆ ∩ 𝑉 = βˆ… . Since 𝐹 ( π‘₯ ) is πœ‡ π‘Œ -connected for each π‘₯ ∈ 𝑋 , we have either 𝐹 ( π‘₯ ) βŠ† π‘ˆ or 𝐹 ( π‘₯ ) βŠ† 𝑉 . If π‘₯ ∈ 𝐹 + ( π‘ˆ βˆͺ 𝑉 ) , then 𝐹 ( π‘₯ ) βŠ† π‘ˆ ∩ 𝑉 and hence π‘₯ ∈ 𝐹 + ( π‘ˆ ) βˆͺ 𝐹 + ( 𝑉 ) . Moreover, since 𝐹 is surjective, there exist π‘₯ and 𝑦 in 𝑋 such that 𝐹 ( π‘₯ ) βŠ† π‘ˆ and 𝐹 ( 𝑦 ) βŠ† 𝑉 ; hence π‘₯ ∈ 𝐹 + ( π‘ˆ ) and 𝑦 ∈ 𝐹 + ( 𝑉 ) . Therefore, we obtain the following:(1) 𝐹 + ( π‘ˆ ) βˆͺ 𝐹 + ( 𝑉 ) = 𝐹 + ( π‘ˆ βˆͺ 𝑉 ) = 𝑋 ,(2) 𝐹 + ( π‘ˆ ) ∩ 𝐹 + ( 𝑉 ) = 𝐹 + ( π‘ˆ ∩ 𝑉 ) = βˆ… ,(3) F + ( π‘ˆ ) β‰  βˆ… and 𝐹 + ( 𝑉 ) β‰  βˆ… . By Theorem 3.3, 𝐹 + ( π‘ˆ ) and 𝐹 + ( 𝑉 ) are πœ‡ 𝑋 - 𝛽 -open. Consequently, ( 𝑋 , πœ‡ 𝑋 ) is not πœ‡ 𝑋 - 𝛽 -connected.

Theorem 3.11. Let 𝐹 ∢ 𝑋 β†’ π‘Œ be lower 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous surjective multifunction. If ( 𝑋 , πœ‡ 𝑋 ) is πœ‡ 𝑋 - 𝛽 -connected and 𝐹 ( π‘₯ ) is πœ‡ π‘Œ - 𝛽 -connected for each π‘₯ ∈ 𝑋 , then ( π‘Œ , πœ‡ π‘Œ ) is πœ‡ π‘Œ - 𝛽 -connected.

Proof. The proof is similar to that of Theorem 3.10 and is thus omitted.

Let { 𝑋 𝛼 ∢ 𝛼 ∈ Ξ¦ } and { π‘Œ 𝛼 ∢ 𝛼 ∈ Ξ¦ } be any two families of generalized topological spaces with the same index set Ξ¦ . For each 𝛼 ∈ Ξ¦ , let 𝐹 𝛼 ∢ 𝑋 𝛼 β†’ π‘Œ 𝛼 be a multifunction. The product space ∏ { 𝑋 𝛼 ∢ 𝛼 ∈ Ξ¦ } is denoted by ∏ 𝑋 𝛼 and the product multifunction ∏ 𝐹 𝛼 ∢ ∏ 𝑋 𝛼 β†’ ∏ π‘Œ 𝛼 , defined by ∏ 𝐹 ( π‘₯ ) = { 𝐹 𝛼 ( π‘₯ 𝛼 ) ∢ 𝛼 ∈ Ξ¦ } for each π‘₯ = { π‘₯ 𝛼 ∏ 𝑋 } ∈ 𝛼 , is simply denoted by ∏ 𝑋 𝐹 ∢ 𝛼 β†’ ∏ π‘Œ 𝛼 .

Theorem 3.12. Let 𝐹 𝛼 ∢ 𝑋 β†’ π‘Œ 𝛼 be a multifunction for each 𝛼 ∈ Ξ¦ and ∏ π‘Œ 𝐹 ∢ 𝑋 β†’ 𝛼 a multifunction defined by ∏ 𝐹 ( π‘₯ ) = { 𝐹 𝛼 ( π‘₯ ) ∢ 𝛼 ∈ Ξ¦ } for each π‘₯ ∈ 𝑋 . If 𝐹 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ ∏ π‘Œ 𝛼 ) -continuous, then 𝐹 𝛼 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ 𝛼 ) -continuous for each 𝛼 ∈ Ξ¦ .

Proof. Let π‘₯ ∈ 𝑋 and 𝛼 ∈ Ξ¦ , and let 𝑉 𝛼 be any πœ‡ π‘Œ 𝛼 -open set of π‘Œ 𝛼 containing 𝐹 𝛼 ( π‘₯ ) . Therefore, we obtain that 𝑝 𝛼 βˆ’ 1 ( 𝑉 𝛼 ) = 𝑉 𝛼 Γ— ∏ { π‘Œ 𝛾 ∢ 𝛾 ∈ Ξ¦ and 𝛾 β‰  𝛼 } is a πœ‡ ∏ π‘Œ 𝛼 -open set of ∏ π‘Œ 𝛼 containing 𝐹 ( π‘₯ ) , where 𝑝 𝛼 is the natural projection of ∏ π‘Œ 𝛼 onto π‘Œ 𝛼 . Since 𝐹 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ ∏ π‘Œ 𝛼 ) -continuous, there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ ) βŠ† 𝑝 𝛼 βˆ’ 1 ( 𝑉 𝛼 ) . Therefore, we obtain 𝐹 𝛼 ( π‘ˆ ) βŠ† 𝑝 𝛼 ( 𝐹 ( π‘ˆ ) ) βŠ† 𝑝 𝛼 ( 𝑝 𝛼 βˆ’ 1 ( 𝑉 𝛼 ) ) = 𝑉 𝛼 . This shows that 𝐹 𝛼 ∢ 𝑋 β†’ π‘Œ 𝛼 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ 𝛼 ) -continuous for each 𝛼 ∈ Ξ¦ .

Theorem 3.13. Let 𝐹 𝛼 ∢ 𝑋 β†’ π‘Œ 𝛼 be a multifunction for each 𝛼 ∈ Ξ¦ and ∏ π‘Œ 𝐹 ∢ 𝑋 β†’ 𝛼 a multifunction defined by ∏ 𝐹 ( π‘₯ ) = { 𝐹 𝛼 ( π‘₯ ) ∢ 𝛼 ∈ Ξ¦ } for each π‘₯ ∈ 𝑋 . If 𝐹 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ ∏ π‘Œ 𝛼 ) -continuous, then 𝐹 𝛼 is upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ 𝛼 ) -continuous for each 𝛼 ∈ Ξ¦ .

Proof. The proof is similar to that of Theorem 3.12 and is thus omitted.

4. Upper and Lower Almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -Continuous Multifunctions

Definition 4.1. Let ( 𝑋 , πœ‡ 𝑋 ) and ( π‘Œ , πœ‡ π‘Œ ) be generalized topological spaces. A multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ is said to be(1)upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous at a point π‘₯ ∈ 𝑋 if, for each πœ‡ π‘Œ -open set 𝑉 of π‘Œ containing 𝐹 ( π‘₯ ) , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ ) βŠ† 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) ,(2)lower almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous at a point π‘₯ ∈ 𝑋 if, for each πœ‡ π‘Œ -open set 𝑉 of π‘Œ such that 𝐹 ( π‘₯ ) ∩ 𝑉 β‰  βˆ… , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( 𝑧 ) ∩ 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) β‰  βˆ… for every 𝑧 ∈ π‘ˆ ,(3)upper almost (resp. lower almost) 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous if 𝐹 has this property at each point of 𝑋 .

Remark 4.2. For a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , the following implication holds: upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous β‡’ upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.
The following example shows that this implication is not reversible.

Example 4.3. Let 𝑋 = { 1 , 2 , 3 , 4 } and π‘Œ = { π‘Ž , 𝑏 , 𝑐 , 𝑑 } . Define a generalized topology πœ‡ 𝑋 = { βˆ… , { 1 } , { 1 , 2 } , { 2 , 3 } , { 1 , 2 , 3 } } on 𝑋 and a generalized topology πœ‡ π‘Œ = { βˆ… , { π‘Ž , 𝑐 } , { 𝑏 , 𝑐 } , { π‘Ž , 𝑏 , 𝑐 } , π‘Œ } on π‘Œ . A multifunction 𝐹 ∢ ( 𝑋 , πœ‡ 𝑋 ) β†’ ( π‘Œ , πœ‡ π‘Œ ) is defined as follows: 𝐹 ( 1 ) = { 𝑏 } , 𝐹 ( 2 ) = 𝐹 ( 4 ) = { 𝑑 } , and 𝐹 ( 3 ) = { 𝑐 } . Then 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous but it is not upper 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.

A subset 𝑁 π‘₯ of a generalized topological space ( 𝑋 , πœ‡ 𝑋 ) is said to be πœ‡ 𝑋 -neighbourhood of a point π‘₯ ∈ 𝑋 if there exists a πœ‡ 𝑋 -open π‘ˆ such that π‘₯ ∈ π‘ˆ βŠ† 𝑁 π‘₯ .

Theorem 4.4. For a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , the following properties are equivalent:(1) 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous at a point π‘₯ ∈ 𝑋 ,(2) π‘₯ ∈ 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) for every πœ‡ π‘Œ -open set 𝑉 of π‘Œ containing 𝐹 ( π‘₯ ) ,(3)for each πœ‡ 𝑋 -open neighbourhood π‘ˆ of π‘₯ and each πœ‡ π‘Œ -open set 𝑉 of π‘Œ containing 𝐹 ( π‘₯ ) , there exists a πœ‡ 𝑋 -open set 𝐺 of 𝑋 such that βˆ… β‰  𝐺 βŠ† π‘ˆ and 𝐺 βŠ† 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ,(4)for each πœ‡ π‘Œ -open set 𝑉 of π‘Œ containing 𝐹 ( π‘₯ ) , there exists π‘ˆ ∈ 𝜎 ( πœ‡ 𝑋 , π‘₯ ) such that π‘ˆ βŠ† 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) .

Proof. ( 1 ) β‡’ ( 2 ) Let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ such that 𝐹 ( π‘₯ ) βŠ† 𝑉 . Then there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) = 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) . Then π‘ˆ βŠ† 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) . Since π‘ˆ is πœ‡ 𝑋 - 𝛽 -open, we have π‘₯ ∈ π‘ˆ βŠ† 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( π‘ˆ ) ) ) βŠ† 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) .
( 2 ) β‡’ ( 3 ) Let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ containing 𝐹 ( π‘₯ ) and π‘ˆ a πœ‡ 𝑋 -open set of 𝑋 containing π‘₯ . Since π‘₯ ∈ 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) , we have π‘ˆ ∩ ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) β‰  βˆ… . Put 𝐺 = π‘ˆ ∩ ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) ; then 𝐺 is a nonempty πœ‡ 𝑋 -open set, 𝐺 βŠ† π‘ˆ ; and 𝐺 βŠ† 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) βŠ† 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) .
( 3 ) β‡’ ( 4 ) Let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ containing 𝐹 ( π‘₯ ) . By πœ‡ 𝑋 ( π‘₯ ) , we denote the family of all πœ‡ 𝑋 -open neighbourhoods of π‘₯ . For each π‘ˆ ∈ πœ‡ 𝑋 ( π‘₯ ) , there exists a πœ‡ 𝑋 -open set 𝐺 π‘ˆ of 𝑋 such that βˆ… β‰  𝐺 π‘ˆ βŠ† π‘ˆ and 𝐺 π‘ˆ βŠ† 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) . Put π‘Š = βˆͺ { 𝐺 π‘ˆ ∢ π‘ˆ ∈ πœ‡ 𝑋 ( π‘₯ ) } ; then π‘Š is a πœ‡ 𝑋 -open set of 𝑋 , π‘₯ ∈ 𝑐 πœ‡ 𝑋 ( π‘Š ) , and π‘Š βŠ† 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) . Moreover, if we put π‘ˆ 0 = π‘Š βˆͺ { π‘₯ } , then we obtain π‘ˆ 0 ∈ 𝜎 ( πœ‡ 𝑋 , π‘₯ ) and π‘ˆ 0 βŠ† 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) .
( 4 ) β‡’ ( 1 ) Let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ containing 𝐹 ( π‘₯ ) . There exists 𝐺 ∈ 𝜎 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐺 βŠ† 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) . Therefore, we obtain π‘₯ ∈ 𝐺 ∩ 𝐹 + ( 𝑉 ) βŠ† 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ∩ ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐺 ) ) ) βŠ† 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ∩ ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) ) = 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) .

Theorem 4.5. For a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , the following properties are equivalent:(1) 𝐹 is lower almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous at a point π‘₯ of 𝑋 ,(2) π‘₯ ∈ 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) for every πœ‡ π‘Œ -open set 𝑉 of π‘Œ such that 𝐹 ( π‘₯ ) ∩ 𝑉 β‰  βˆ… ,(3)for any πœ‡ 𝑋 -open neighbourhood π‘ˆ of π‘₯ and a πœ‡ π‘Œ -open set 𝑉 of π‘Œ such that 𝐹 ( π‘₯ ) ∩ 𝑉 β‰  βˆ… , there exists a nonempty πœ‡ 𝑋 -open set 𝐺 of 𝑋 such that 𝐺 βŠ† π‘ˆ and 𝐺 βŠ† 𝑐 πœ‡ ( 𝐹 βˆ’ ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ,(4)for any πœ‡ π‘Œ -open set 𝑉 of π‘Œ such that 𝐹 ( π‘₯ ) ∩ 𝑉 β‰  βˆ… , there exists π‘ˆ ∈ 𝜎 ( πœ‡ 𝑋 , π‘₯ ) such that π‘ˆ βŠ† 𝑐 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) .

Proof. The proof is similar to that of Theorem 4.4 and is thus omitted.

Theorem 4.6. For a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ , the following properties are equivalent:(1) 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(2)for each π‘₯ ∈ 𝑋 and each πœ‡ π‘Œ -open set 𝑉 of π‘Œ containing 𝐹 ( π‘₯ ) , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) ,(3)for each π‘₯ ∈ 𝑋 and each πœ‡ π‘Œ π‘Ÿ -open set 𝑉 of π‘Œ containing 𝐹 ( π‘₯ ) , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ ) βŠ† 𝑉 ,(4) 𝐹 + ( 𝑉 ) ∈ 𝛽 ( πœ‡ 𝑋 ) for every πœ‡ π‘Œ π‘Ÿ -open set 𝑉 of π‘Œ ,(5) 𝐹 βˆ’ ( 𝑀 ) is πœ‡ 𝑋 - 𝛽 -closed in 𝑋 for every πœ‡ π‘Œ π‘Ÿ -closed set 𝑀 of π‘Œ ,(6) 𝐹 + ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) for every πœ‡ π‘Œ -open set 𝑉 of π‘Œ ,(7) 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑖 𝜎 π‘Œ ( 𝑀 ) ) ) βŠ† 𝐹 βˆ’ ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(8) 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) ) ) βŠ† 𝐹 βˆ’ ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(9) 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝐴 ) ) ) ) ) βŠ† 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝐴 ) ) for every subset 𝐴 of π‘Œ ,(10) 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) ) ) ) ) βŠ† 𝐹 βˆ’ ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(11) 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑖 𝜎 π‘Œ ( 𝑀 ) ) ) ) ) βŠ† 𝐹 βˆ’ ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(12) 𝐹 + ( 𝑉 ) βŠ† 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) for every πœ‡ π‘Œ -open set 𝑉 of π‘Œ .

Proof. ( 1 ) β‡’ ( 2 ) The proof follows immediately from Definition 4.1(1).
( 2 ) β‡’ ( 3 ) This is obvious.
( 3 ) β‡’ ( 4 ) Let 𝑉 be any πœ‡ π‘Œ π‘Ÿ -open set of π‘Œ and π‘₯ ∈ 𝐹 + ( 𝑉 ) . Then 𝐹 ( π‘₯ ) βŠ† 𝑉 and there exists π‘ˆ π‘₯ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ π‘₯ ) βŠ† 𝑉 . Therefore, we have π‘₯ ∈ π‘ˆ π‘₯ βŠ† 𝐹 + ( 𝑉 ) and hence 𝐹 + ( 𝑉 ) ∈ 𝛽 ( πœ‡ 𝑋 ) .
( 4 ) β‡’ ( 5 ) This follows from the fact that 𝐹 + ( π‘Œ βˆ’ 𝑀 ) = 𝑋 βˆ’ 𝐹 βˆ’ ( 𝑀 ) for every subset 𝑀 of π‘Œ .
( 5 ) β‡’ ( 6 ) Let 𝑉 be any πœ‡ 𝑋 -open set of π‘Œ and π‘₯ ∈ 𝐹 + ( 𝑉 ) . Then we have 𝐹 ( π‘₯ ) βŠ† 𝑉 βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) and hence π‘₯ ∈ 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) = 𝑋 βˆ’ 𝐹 βˆ’ ( π‘Œ βˆ’ 𝑐 𝜎 π‘Œ ( 𝑉 ) ) . Since π‘Œ βˆ’ 𝑐 𝜎 π‘Œ ( 𝑉 ) is πœ‡ π‘Œ π‘Ÿ -closed set of π‘Œ , 𝐹 βˆ’ ( π‘Œ βˆ’ 𝑐 𝜎 π‘Œ ( 𝑉 ) ) is πœ‡ 𝑋 - 𝛽 -closed in 𝑋 . Therefore, 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) and hence π‘₯ ∈ 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) . Consequently, we obtain 𝐹 + ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) .
( 6 ) β‡’ ( 7 ) Let 𝑀 be any πœ‡ π‘Œ -closed set of π‘Œ . Then, since π‘Œ βˆ’ 𝑀 is πœ‡ π‘Œ -open, we obtain 𝑋 βˆ’ 𝐹 βˆ’ ( 𝑀 ) = 𝐹 + ( π‘Œ βˆ’ 𝑀 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( π‘Œ βˆ’ 𝑀 ) ) ) = 𝑖 𝛽 𝑋 ( 𝐹 + ( π‘Œ βˆ’ 𝑖 𝜎 π‘Œ ( 𝐾 ) ) ) = 𝑖 𝛽 𝑋 ( 𝑋 βˆ’ 𝐹 βˆ’ ( 𝑖 𝜎 π‘Œ 𝑐 ( 𝑀 ) ) ) = 𝑋 βˆ’ 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑖 𝜎 π‘Œ ( 𝑀 ) ) ) . Therefore, we obtain 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑖 𝜎 π‘Œ ( 𝑀 ) ) ) βŠ† 𝐹 βˆ’ ( 𝑀 ) .
( 7 ) β‡’ ( 8 ) The proof is obvious since 𝑖 𝜎 π‘Œ ( 𝑀 ) = 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) for every πœ‡ π‘Œ -closed set 𝑀 .
( 8 ) β‡’ ( 9 ) The proof is obvious.
( 9 ) β‡’ ( 1 0 ) Since 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝐴 ) ) ) βŠ† 𝑐 𝛽 π‘Œ ( 𝐴 ) for every subset 𝐴 , for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ , we have 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) ) ) ) ) βŠ† 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) ) ) = 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑀 ) ) ) ) ) βŠ† 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑀 ) ) = 𝐹 βˆ’ ( 𝑀 ) .
( 1 0 ) β‡’ ( 1 1 ) The proof is obvious since 𝑖 𝜎 π‘Œ ( 𝑀 ) = 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) for every πœ‡ 𝑋 -closed set 𝑀 .
( 1 1 ) β‡’ ( 1 2 ) Let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ . Then π‘Œ βˆ’ 𝑉 is πœ‡ π‘Œ -closed in π‘Œ and we have 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑖 𝜎 π‘Œ ( π‘Œ βˆ’ 𝑉 ) ) ) ) ) βŠ† 𝐹 βˆ’ ( π‘Œ βˆ’ 𝑉 ) = 𝑋 βˆ’ 𝐹 + ( 𝑉 ) . Moreover, we have 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑖 𝜎 π‘Œ ( π‘Œ βˆ’ 𝑉 ) ) ) ) ) = 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 βˆ’ ( π‘Œ βˆ’ 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) = 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑋 βˆ’ 𝐹 + ( 𝑐 𝜎 π‘Œ 𝑐 ( 𝑉 ) ) ) ) ) = 𝑋 βˆ’ πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) . Therefore, we obtain 𝐹 + ( 𝑉 ) βŠ† 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) .
( 1 2 ) β‡’ ( 1 ) Let π‘₯ be any point of 𝑋 and 𝑉 any πœ‡ π‘Œ -open set of π‘Œ containing 𝐹 ( π‘₯ ) . Then π‘₯ ∈ 𝐹 + ( 𝑉 ) βŠ† 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) and hence 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous at π‘₯ by Theorem 4.4.

Theorem 4.7. The following are equivalent for a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ :(1) 𝐹 is lower almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(2)for each π‘₯ ∈ 𝑋 and each πœ‡ π‘Œ -open set 𝑉 of π‘Œ such that 𝐹 ( π‘₯ ) ∩ 𝑉 β‰  βˆ… , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that π‘ˆ βŠ† 𝐹 βˆ’ ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ,(3)for each π‘₯ ∈ 𝑋 and each πœ‡ π‘Œ π‘Ÿ -open set 𝑉 of π‘Œ such that 𝐹 ( π‘₯ ) ∩ 𝑉 β‰  βˆ… , there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that π‘ˆ βŠ† 𝐹 βˆ’ ( 𝑉 ) ,(4) 𝐹 βˆ’ ( 𝑉 ) ∈ 𝛽 ( πœ‡ 𝑋 ) for every πœ‡ π‘Œ π‘Ÿ -open set 𝑉 of π‘Œ ,(5) 𝐹 + ( 𝑀 ) is πœ‡ 𝑋 - 𝛽 -closed in 𝑋 for every πœ‡ π‘Œ π‘Ÿ -closed set 𝑀 of π‘Œ ,(6) 𝐹 βˆ’ ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) for every πœ‡ π‘Œ -open set 𝑉 of π‘Œ ,(7) 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝑖 𝜎 π‘Œ ( 𝑀 ) ) ) βŠ† 𝐹 + ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(8) 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) ) ) βŠ† 𝐹 + ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(9) 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝐴 ) ) ) ) ) βŠ† 𝐹 + ( 𝑐 πœ‡ π‘Œ ( 𝐴 ) ) for every subset 𝐴 of π‘Œ ,(10) 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 + ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( 𝑀 ) ) ) ) ) ) βŠ† 𝐹 + ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(11) 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝐹 + ( 𝑖 𝜎 π‘Œ ( 𝑀 ) ) ) ) ) βŠ† 𝐹 + ( 𝑀 ) for every πœ‡ π‘Œ -closed set 𝑀 of π‘Œ ,(12) 𝐹 βˆ’ ( 𝑉 ) βŠ† 𝑐 πœ‡ 𝑋 ( 𝑖 πœ‡ 𝑋 ( 𝑐 πœ‡ 𝑋 ( 𝐹 βˆ’ ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) ) ) ) for every πœ‡ π‘Œ -open set 𝑉 of π‘Œ .

Proof. The proof is similar to that of Theorem 4.6 and is thus omitted.

Theorem 4.8. The following are equivalent for a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ :(1) 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(2) 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑉 ) ) βŠ† 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) for every 𝑉 ∈ 𝛽 ( πœ‡ π‘Œ ) ,(3) 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑉 ) ) βŠ† 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) for every 𝑉 ∈ 𝜎 ( πœ‡ π‘Œ ) ,(4) 𝐹 + ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) ) ) for every 𝑉 ∈ πœ‹ ( πœ‡ π‘Œ ) .

Proof. ( 1 ) β‡’ ( 2 ) Let 𝑉 be any πœ‡ π‘Œ - 𝛽 -open set of π‘Œ . Since 𝑐 πœ‡ π‘Œ ( 𝑉 ) is πœ‡ π‘Œ π‘Ÿ -closed, by Theorem 4.6   𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) is πœ‡ 𝑋 - 𝛽 -closed in 𝑋 and 𝐹 βˆ’ ( 𝑉 ) βŠ† 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) . Therefore, we obtain 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑉 ) ) βŠ† 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) .
( 2 ) β‡’ ( 3 ) This is obvious since 𝜎 ( πœ‡ π‘Œ ) βŠ† 𝛽 ( πœ‡ π‘Œ ) .
( 3 ) β‡’ ( 4 ) Let 𝑉 ∈ πœ‹ ( πœ‡ π‘Œ ) . Then, we have 𝑉 βŠ† 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) and π‘Œ βˆ’ 𝑉 βŠ‡ 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( π‘Œ βˆ’ 𝑉 ) ) . Since 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( π‘Œ βˆ’ 𝑉 ) ) ∈ 𝜎 ( πœ‡ π‘Œ ) , we have 𝑋 βˆ’ 𝐹 + ( 𝑉 ) = 𝐹 βˆ’ ( π‘Œ βˆ’ 𝑉 ) βŠ‡ 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( π‘Œ βˆ’ 𝑉 ) ) ) βŠ‡ 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑐 πœ‡ π‘Œ ( 𝑖 πœ‡ π‘Œ ( π‘Œ βˆ’ 𝑉 ) ) ) ) = 𝑐 𝛽 𝑋 ( 𝐹 βˆ’ ( π‘Œ βˆ’ 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ 𝑐 ( 𝑉 ) ) ) ) = 𝛽 𝑋 ( 𝑋 βˆ’ 𝐹 + ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) ) ) = 𝑋 βˆ’ 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) ) ) . Therefore, we obtain 𝐹 + ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) ) ) .
( 4 ) β‡’ ( 1 ) Let 𝑉 be any πœ‡ π‘Œ π‘Ÿ -open set of π‘Œ . Since 𝑉 ∈ πœ‹ ( πœ‡ π‘Œ ) , we have 𝐹 + ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) ) ) = 𝑖 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) and hence 𝐹 + ( 𝑉 ) ∈ 𝛽 ( πœ‡ 𝑋 ) . It follows from Theorem 4.6 that 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.

Theorem 4.9. The following are equivalent for a multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ :(1) 𝐹 is lower almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(2) 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) βŠ† 𝐹 + ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) for every 𝑉 ∈ 𝛽 ( πœ‡ π‘Œ ) ,(3) 𝑐 𝛽 𝑋 ( 𝐹 + ( 𝑉 ) ) βŠ† 𝐹 + ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) for every 𝑉 ∈ 𝜎 ( πœ‡ π‘Œ ) ,(4) 𝐹 βˆ’ ( 𝑉 ) βŠ† 𝑖 𝛽 𝑋 ( 𝐹 βˆ’ ( 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) ) ) for every 𝑉 ∈ πœ‹ ( πœ‡ π‘Œ ) .

Proof. The proof is similar to that of Theorem 4.8 and is thus omitted.

For a multifunction 𝑋 β†’ π‘Œ , by 𝑐 πœ‡ 𝐹 ∢ 𝑋 β†’ π‘Œ we denote a multifunction defined as follows: ( 𝑐 πœ‡ 𝐹 ) ( π‘₯ ) = 𝑐 πœ‡ π‘Œ ( 𝐹 ( π‘₯ ) ) for each π‘₯ ∈ 𝑋 . Similarly, we can define 𝑐 𝛽 𝐹 ∢ 𝑋 β†’ π‘Œ , 𝑐 𝜎 𝐹 ∢ 𝑋 β†’ π‘Œ , 𝑐 πœ‹ 𝐹 ∢ 𝑋 β†’ π‘Œ , and 𝑐 𝛼 𝐹 ∢ 𝑋 β†’ π‘Œ .

Theorem 4.10. A multifunction 𝐹 ∢ 𝑋 β†’ π‘Œ is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous if and only if 𝑐 𝜎 𝐹 ∢ 𝑋 β†’ π‘Œ is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.

Proof. Suppose that 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous. Let π‘₯ ∈ 𝑋 , and let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ such that ( 𝑐 𝜎 𝐹 ) ( π‘₯ ) βŠ† 𝑉 . Then 𝐹 ( π‘₯ ) βŠ† 𝑉 and by Theorem 4.6 there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that 𝐹 ( π‘ˆ ) βŠ† 𝑐 𝛽 π‘Œ ( 𝑉 ) . For each 𝑒 ∈ π‘ˆ , 𝐹 ( 𝑒 ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) and hence 𝑐 𝜎 π‘Œ ( 𝐹 ( π‘ˆ ) ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) . Therefore, we have ( 𝑐 𝜎 𝐹 ) ( π‘ˆ ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) and by Theorem 4.6   𝑐 𝜎 𝐹 is is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.
Conversely, suppose that 𝑐 𝜎 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous. Let π‘₯ ∈ 𝑋 , and let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ containing 𝐹 ( π‘₯ ) . Then 𝐹 ( π‘₯ ) βŠ† 𝑉 and 𝑐 𝜎 π‘Œ ( 𝐹 ( π‘₯ ) ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) . Since 𝑐 𝜎 π‘Œ ( 𝑉 ) = 𝑖 πœ‡ π‘Œ ( 𝑐 πœ‡ π‘Œ ( 𝑉 ) ) is πœ‡ π‘Œ -open, there exists π‘ˆ ∈ 𝛽 ( πœ‡ 𝑋 , π‘₯ ) such that ( 𝑐 𝜎 𝐹 ) ( π‘ˆ ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑐 𝜎 π‘Œ ( 𝑉 ) ) = 𝑐 𝜎 π‘Œ ( 𝑉 ) . Therefore, we have 𝐹 ( π‘ˆ ) βŠ† 𝑐 𝜎 π‘Œ ( 𝑉 ) and hence 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.

Definition 4.11. A subset 𝐴 of a generalized topological space ( 𝑋 , πœ‡ 𝑋 ) is said to be πœ‡ 𝑋 - 𝛼 -paracompact if every cover of 𝐴 by πœ‡ 𝑋 -open sets of 𝑋 is refined by a cover of 𝐴 that consists of πœ‡ 𝑋 -open sets of 𝑋 and is locally finite in 𝑋 .

Definition 4.12. A subset 𝐴 of a generalized topological space ( 𝑋 , πœ‡ 𝑋 ) is said to be πœ‡ 𝑋 - 𝛼 -regular if, for each point π‘₯ ∈ 𝐴 and each πœ‡ 𝑋 -open set π‘ˆ of 𝑋 containing π‘₯ , there exists a πœ‡ 𝑋 -open set 𝐺 of 𝑋 such that π‘₯ ∈ 𝐺 βŠ† 𝑐 πœ‡ 𝑋 ( 𝐺 ) βŠ† π‘ˆ .

Lemma 4.13. If 𝐴 is a πœ‡ 𝑋 - 𝛼 -regular πœ‡ 𝑋 - 𝛼 -paracompact subset of a quasitopological space ( 𝑋 , πœ‡ 𝑋 ) and π‘ˆ is a πœ‡ 𝑋 -open neighbourhood of 𝐴 , then there exists a πœ‡ 𝑋 -open set 𝐺 of 𝑋 such that 𝐴 βŠ† 𝐺 βŠ† 𝑐 πœ‡ 𝑋 ( 𝐺 ) βŠ† π‘ˆ .

Lemma 4.14. Let ( 𝑋 , πœ‡ 𝑋 ) be a generalized topological space and ( π‘Œ , πœ‡ π‘Œ ) a quasitopological space. If 𝐹 ∢ 𝑋 β†’ π‘Œ is a multifunction such that 𝐹 ( π‘₯ ) is πœ‡ π‘Œ - 𝛼 -paracompact πœ‡ π‘Œ - 𝛼 -regular for each π‘₯ ∈ 𝑋 , then for each πœ‡ π‘Œ -open set 𝑉 of π‘Œ    𝐺 + ( 𝑉 ) = 𝐹 + ( 𝑉 ) , where 𝐺 denotes 𝑐 𝛽 𝐹 , 𝑐 πœ‹ 𝐹 , 𝑐 𝛼 𝐹 , or 𝑐 πœ‡ 𝐹 .

Proof. Let 𝑉 be any πœ‡ π‘Œ -open set of π‘Œ and π‘₯ ∈ 𝐺 + ( 𝑉 ) . Thus 𝐺 ( π‘₯ ) βŠ† 𝑉 and 𝐹 ( π‘₯ ) βŠ† 𝐺 ( π‘₯ ) βŠ† 𝑉 . We have π‘₯ ∈ 𝐹 + ( 𝑉 ) and hence 𝐺 + ( 𝑉 ) βŠ† 𝐹 + ( 𝑉 ) . Let π‘₯ ∈ 𝐹 + ( 𝑉 ) ; then 𝐹 ( π‘₯ ) βŠ† 𝑉 . By Lemma 4.13, there exists a πœ‡ π‘Œ -open set π‘Š of π‘Œ such that 𝐹 ( π‘₯ ) βŠ† π‘Š βŠ† 𝑐 πœ‡ π‘Œ ( π‘Š ) βŠ† 𝑉 ; hence 𝐺 ( π‘₯ ) βŠ† 𝑐 πœ‡ π‘Œ ( π‘Š ) βŠ† 𝑉 . Therefore, we have π‘₯ ∈ 𝐺 + ( 𝑉 ) and 𝐹 + ( 𝑉 ) βŠ† 𝐺 + ( 𝑉 ) .

Theorem 4.15. Let ( 𝑋 , πœ‡ 𝑋 ) be a generalized topological space and ( π‘Œ , πœ‡ π‘Œ ) a quasitopological space. Let 𝐹 ∢ 𝑋 β†’ π‘Œ be a multifunction such that 𝐹 ( π‘₯ ) is πœ‡ π‘Œ - 𝛼 -paracompact and πœ‡ π‘Œ - 𝛼 -regular for each π‘₯ ∈ 𝑋 . Then the following are equivalent:(1) 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(2) 𝑐 𝛽 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(3) 𝑐 πœ‹ 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(4) 𝑐 𝛼 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous,(5) 𝑐 πœ‡ 𝐹 is upper almost 𝛽 ( πœ‡ 𝑋 , πœ‡ π‘Œ ) -continuous.

Proof. Similarly to Lemma 4.14, we put 𝐺 = 𝑐 𝛽 𝐹 , 𝑐 πœ‹ 𝐹 , 𝑐 𝛼 𝐹 , or