pj-b-B-continuous, pj-b-t-continuous, pj-b-semi-continuous) if is pj-b-preopen (resp. Afunction is called pj-b-pre-con- tinuous (resp. It is following from lemma 3.4 in ĭefinition 3.3. A function is called pj-B-conti- nuous if and only if it is locally pj-b-closed-continuous and pj-semi-continuous. pj-Locally b-closed set, pj-D(c,b)-set, pj-α-open, pj-semiopen, jp-semiopen, pj-B-set, pj-Locally closed, jp-rgular) in for each p-open set V of Y. pj-Locally b-closed continuous, pj-D(c,b)-continuous, pj-α-continuous pj-semi continuous, jp-semi continuous, pj-B- continuous, pj-Locally closed continuous, jp-regular continuous ) if is pj-b-set (resp. A function is called pj-b-conti- nuous (resp. Decompositions of New Kinds of ContinuityĪfter we had been defined and studied the propriety of our new classes of sets we are ready to study the concept of continuity between any two bitopological spaces via our new classes of sets.ĭefinition 3.1. A subset of a bitopological space is jp-regular open if and only if it is pj-α-open and pj-b-t-set.ģ. Therefore by Proposition 2.3 A is jp-regular open.Ĭorollary 2.17. (2) Þ (1) Let be pj-b-preopen and pj-sb-generalized closed. (1) Þ (2) Let be jp-regular open.Then is pj-b-open. Let be a subset of a bitopological space, the following properties are equivalent:Ģ) is pj-b-preopen and pj-sb-generalized closed set. pj- is the intersection of all pj-semiclosed sets which containing. A subset of a bitopological space is called pj-sb- generalized closed if pj-, whenever and is pj-b- preopen.ĭefinition 2.15. (3) Þ (1) Let be pj-b-preopen and pj-b-t-set.Then and is p-open by lemma 2.1 Hence, is jp-regular set.ĭefinition 2.14. (1) Þ (2) Let be jp-regular set.since then. Let be a subsets of a bitopological space, then the following are equivalent: A subset of a bitopological space is p-open if and only if it is pj-α-open and pj-b-B-set. From example 2.4 it is clear that it is p1-b-preopen but it is not a p1-b-B-set.Ĭorollary 2.12. From example 2.3 it is clear that is a p1-b-B -set but it is not p1-b-preopen.Įxample 2.11. The following examples show that pj-b-preopen sets and pj-b-B-sets are independent.Įxample 2.10. (2) Þ (1) be pj-b-preopen and pj-b-B-set. (1) Þ (2) Let be p-open but then is pj-b-preopen. Let be a subset of a bitopological space, then the following are equivalent: , where is p-open and is a pj-t- set i.e. and is p-open set, then is pj-b- B-set.ģ) Let be pj-B-set i.e. Hence is pj-b-t-set.Ģ) Let be pj-b-t-set. 1) Let be pj-t-set,then from lem- ma 2.1. Let be a subsets of a bitopological space, thenĢ) If is pj-b-t-set then it is pj-b-B-set. Let be p-open subset of a bitopological space, then From example 2.2 it is clear that is a p2-b-t-set but it is not p2-b-closed. The following example shows that the converse of (2) is not true in general.Įxample 2.6. conversely, Let be pj-b-semiclosed set, then. 1) Let be pj-b-t set, then that implies is pj-b-semiclosed. If and are a subsets of a bitopological space, thenġ) is a pj-b-t set if and only if is pj-b-semiclosed.Ģ) If is pj-b-closed, then it is a pj-b-t-set.ģ) If and are pj-b-t-sets, then is a pj-b-t-set. Let and and then is a p1-b-B-set.Įxample 2.4. Let, and then is a p2-b-t-set.Įxample 2.3. A subset of a bitopological space is said to be:Ģ) pj-b-B-set if, where is p-open and is a pj-b-t-set.Įxample 2.2. In this section, we investigated our new classes of sets pj-b-preopen, pj-b- semiopen, pj-b-t set, pj-b-B set and pj-sb-generalized closed set and study some of its fundamental properties and examples also we introduce some of important theories which is useful to study the decomposition of continuity via our new classes of sets.ĭefinition 2.1. pj-b-t-Set, pj-b-B-Set pj-b-Semiopen, pj-b-Preopen and pj-sb-Generalized Closed A subset of a bitopological space is said to be:Ħ) pj-B-set if, where is p-open and is a pj-t-set.Ģ. Also, the p-closure of (or ) is the intersection of all p-closed sets which containing. p-interior of (or ) is the union of all p-open sets of a bitopological space which contained in a subset of. p-closed is the com- plement of p-open set. Let be a subset of a bitopological space then called pairwise p-open (or p-open) if. Let be a subset of a space, then is said to be:Ģ) b-B-set if, where and is a b-t-set.ģ) Locally b-closed if, where and is a b-closed set.ĭefinition 1.2. In this paper, we introduce decomposition of continuity in bitopological space via new classes of sets called pj-b-preopen, pj-b-B set, pj-b-t set, pj-b-semi-open and pj-sb-genera- lized closed set with some theories, examples and results.ĭefinition 1.1. Decomposition of pair- wise continuity was given by Jelice and. Tong introduced the concept of t-set and B-set in topological space. In topological space, there are many classes of generalized open sets given by.
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