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International Conference on Systems, Science, Control, Communication, Engineering and Technology 2015 [ICSSCCET 2015]
ISBN Website Received Article ID
978-81-929866-1-6 icssccet.org 10 - July - 2015 ICSSCCET036
VOL eMail Accepted eAID
01 icssccet@asdf.res.in 31- July - 2015 ICSSCCET.2015.036
STRONGLY g*-CLOSED SETS IN BITOPOLOGICAL SPACES J. Logeshwari1, J. Manonmani2, S. Padmanaban3, S. Gowri Sankar4 1, 2, 3, 4
Assistant Professors, Department of Mathematics, Karpagam Institute of Technology, Coimbatore.
Abstract: The purpose of this paper is to define and study đ?‘ đ?‘”∗ -closed sets in bitopological spaces Keywords: Strongly đ?‘”∗ -closed sets
1. INTRODUCTION AND PRELIMINARIES A triple (đ?‘‹, đ?œ?1, đ?œ?2 ) where đ?‘‹ is a non-empty set and đ?œ?1 and đ?œ?2 are topologies on đ?‘‹ is called a bitopological space and Kelly [3] initiated the study of such spaces. Throughout this paper (đ?‘‹, đ?œ?1 , đ?œ?2 ) or simply đ?‘‹ represents the bitopological spaces on which no seperaxion axioms are assumed unless otherwise mentioned. For any subset đ??´ ⊆ đ?‘‹, đ?œ?đ?‘– -đ?‘–đ?‘›đ?‘Ą(đ??´) and , đ?œ?đ?‘– -đ?‘?đ?‘™(đ??´) denote the interior and closure of a set đ??´ with respect to the topology đ?œ?đ?‘– , respectively. Definition 1.1: A subset đ??´ of a topological space (đ?‘‹, đ?œ?) is said to be (1) Semi-open [4 ] if đ??´ ⊆ đ?‘?đ?‘™(đ?‘–đ?‘›đ?‘Ą(đ??´)) (2) Regular open if đ??´ = đ?‘–đ?‘›đ?‘Ą(đ?‘?đ?‘™(đ??´)) Definition 1.2: A subset đ??´ of a topological space (đ?‘‹, đ?œ?) is said to be generalized closed (briefly g-closed) [1 ] if đ?‘?đ?‘™(đ??´) ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ and đ?‘ˆ is open in đ?‘‹. Definition 1.3: A subset đ??´ of a topological space (đ?‘‹, đ?œ?) is said to be generalized* closed (briefly đ?’ˆâˆ— -closed) [6] if đ?‘?đ?‘™(đ??´) ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ and đ?‘ˆ is đ?‘”-open in đ?‘‹. Definition 1.4: A subset đ??´ of a bitopological space (đ?‘‹, đ?œ?1 , đ?œ?2 ) is said to be (1) đ?œ?1 đ?œ?1 -Semi-open [2 ] if đ??´ ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) (2) đ?œ?1 đ?œ?2 -Regular open if đ??´ = đ?‘–đ?‘›đ?‘Ą(đ?‘?đ?‘™(đ??´)) Definition 1.5: A subset đ??´ of a bitopological space (đ?‘‹, đ?œ?1 , đ?œ?2 ) is said to be đ?œ?1 đ?œ?2 -generalized closed (đ?œ?1 đ?œ?2 -g- closed) [ ] if đ?œ?2 đ?‘?đ?‘™(đ??´) ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ and đ?‘ˆ is đ?œ?1 -open. Definition 1.6: A subset đ??´ of a bitopological space (đ?‘‹, đ?œ?1 , đ?œ?2 ) is said to be đ?œ?1 đ?œ?2 -generalized* closed (đ?œ?1 đ?œ?2 -đ?‘”∗ - closed) [ ] if đ?œ?2 − đ?‘?đ?‘™(đ??´) ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ and đ?‘ˆ is đ?œ?1 -đ?‘” open. 2. STRONGLY g*-CLOSED SETS This paper is prepared exclusively for International Conference on Systems, Science, Control, Communication, Engineering and Technology 2015 [ICSSCCET] which is published by ASDF International, Registered in London, United Kingdom. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honoured. For all other uses, contact the owner/author(s). Copyright Holder can be reached at copy@asdf.international for distribution.
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Cite this article as: J Logeshwari, J Manonmani, S Padmanaban, S Gowri Sankar. “STRONGLY g*-CLOSED SETS IN BITOPOLOGICAL SPACES.� International Conference on Systems, Science, Control, Communication, Engineering and Technology (2015): 167-169. Print.
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Definition 2.1: Let (đ?‘‹, đ?œ?1 , đ?œ?2 ) be a bitopological space and đ??´ be its subset, then A is a strongly đ?‘”∗ -closed set (briefly đ?‘ đ?‘”∗ -closed) if đ?œ?2 -đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ and đ?‘ˆ is đ?œ?1 - đ?‘” open Theorem 2.2: Let (đ?‘‹, đ?œ?1 , đ?œ?2 ) be a bitopological space. Every closed set is strongly đ?‘”∗ -closed set, but not conversely. Proof: Suppose that đ??´ is closed. Let đ?‘ˆ be an open set containing đ??´. Then đ?œ?2 -đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊆ đ?œ?2 − đ?‘?đ?‘™(đ??´) = đ??´, which implies, đ?œ?2 -đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊆ đ?‘ˆ. Hence, đ??´ is a strongly đ?‘”∗ -closed set. Example 2.3: Let đ?‘‹ = {đ?‘Ž, đ?‘?, đ?‘?}, đ?œ?1 = {∅, {a}, {b}, {a, b}, X} and đ?œ?2 = {∅, {đ?‘Ž}, {đ?‘Ž, đ?‘?}, đ?‘‹}. Then the set {đ?‘Ž, đ?‘?} is a strongly đ?‘”∗ closed set but not a closed set. Theorem 2.4: If a subset đ??´ of a bitopological space (đ?‘‹, đ?œ?1 , đ?œ?2 ) is đ?‘”∗ -closed then it is strongly đ?‘”∗ -closed in đ?‘‹, but not conversely. Proof: Suppose đ??´ is đ?‘”∗ -closed in (đ?‘‹, đ?œ?1 , đ?œ?2 ). Let đ??ş be an open set containing đ??´ in đ?‘‹. Then đ??ş contains đ?œ?2 − đ?‘?đ?‘™(đ??´) and đ??ş ⊇ đ?œ?2 − đ?‘?đ?‘™(đ??´) ⊇ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)). Thus, đ??´ is strongly đ?‘”∗ -closed in đ?‘‹. Example 2.5: Let đ?‘‹ = {đ?‘Ž, đ?‘?, đ?‘?} with topologies đ?œ?1 = {∅, {đ?‘?}, {đ?‘?}, {đ?‘?, đ?‘?}, đ?‘‹} and đ?œ?2 = {∅, {đ?‘Ž, đ?‘?}, đ?‘‹}. In this topological space the subset {đ?‘Ž} is strongly đ?‘”∗ -closed but not a đ?‘”∗ -closed set. Theorem 2.6: If a subset đ??´ of a topological space (đ?‘‹, đ?œ?1 , đ?œ?2 ) is both open and strongly đ?‘”∗ -closed, then it is closed. Proof: Suppose a subset đ??´ of đ?‘‹ is both open and strongly đ?‘”∗ -closed. Then đ??´ ⊇ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊇ đ?œ?2 − đ?‘?đ?‘™(đ??´) and so đ??´ ⊇ đ?œ?2 − đ?‘?đ?‘™(đ??´). Since đ?œ?2 − đ?‘?đ?‘™(đ??´) ⊇ đ??´, we have, đ??´ = đ?œ?2 − đ?‘?đ?‘™(đ??´). Thus đ??´ is closed in đ?‘‹. Theorem 2.7: If a subset đ??´ of a bitopological space (đ?‘‹, đ?œ?1 , đ?œ?2 ) is both strongly đ?‘”∗ -closed and semi-open then it is đ?‘”∗ -closed. Proof: Suppose đ??´ is both strongly đ?‘”∗ -closed and semi-open in đ?‘‹, let đ??ş be an open set containing đ??´. As đ??´ is strongly đ?‘”∗ -closed, đ??ş ⊇ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)). Now, đ??ş ⊇ đ?œ?2 − đ?‘?đ?‘™(đ??´), since đ??´ is semi-open. Thus đ??´ is đ?‘”∗ -closed in đ?‘‹. Corollary 2.8: If a subset đ??´ of a bitopological space (đ?‘‹, đ?œ?1 , đ?œ?2 ) is both strongly đ?‘”∗ -closed and open then it is a đ?‘”∗ -closed set. Proof: Suppose đ??´ is both strongly đ?‘”∗ -closed and open in đ?‘‹, let đ??ş be an open set containing đ??´. As đ??´ is strongly đ?‘”∗ -closed, đ??ş ⊇ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) and đ??ş ⊇ đ?œ?2 − đ?‘?đ?‘™(đ??´), since đ??´ is open. Thus, đ??´ is đ?‘”∗ -closed in đ?‘‹. Theorem 2.9: A subset đ??´ is strongly đ?‘”∗ -closed if and only if đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ contains no non-empty closed set. Necessity: Suppose that đ??š is a non-empty closed subset of đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´. i.e., đ??š ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ∊ (đ?‘‹ − đ??´). Then đ??š ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) and đ??š ⊆ (đ?‘‹ − đ??´). Since đ?‘‹ − đ??š is an open set and đ??´ is strongly đ?‘”∗ -closed, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊆ (đ?‘‹ − đ??š). i.e., đ??š ⊆ (đ?‘‹ − đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´))) . Hence, đ??š ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ∊ (đ?‘‹ − (đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) = ∅. i.e., đ??š = ∅. Thus, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ contains no non-empty closed set. Sufficiency: Conversely, assume that đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ contains no non-empty closed set. Let đ??´ ⊆ đ?‘ˆ, đ?‘ˆ is g-open. Suppose that đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) is not contained in đ?‘ˆ. Then đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ∊ (đ?‘‹ − đ?‘ˆ) is a non-empty closed set and contained in đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ which is a contradiction. Therefore, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊆ đ?‘ˆ and hence A is strongly đ?‘”∗ - closed. Corollary 2.10: A strongly đ?‘”∗ - closed set A is regular closed if and only if đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ is closed and đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊇ đ??´. Proof: Assume that A is regular closed. Since đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) = đ??´, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ = ∅ is regular closed and hence closed. Conversely, assume that đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ is closed. By Theorem 9, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ contains no non-empty closed set. Therefore, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ = ∅. Thus, A is regular closed. Theorem 2.11: Suppose that đ??ľ ⊆ đ??´ ⊆ đ?‘‹, B is a strongly đ?‘”∗ - closed set relative to A and that both open and strongly đ?‘”∗ -closed subset of (đ?‘‹, đ?œ?1 , đ?œ?2 ) then B is a strongly đ?‘”∗ - closed set relative to (đ?‘‹, đ?œ?1 , đ?œ?2 ). Proof : Let đ??ľ ⊆ đ??ş and G be an open set in (đ?‘‹, đ?œ?1 , đ?œ?2 ). But given that đ??ľ ⊆ đ??´ ⊆ đ?‘‹, therefore đ??ľ ⊆ đ??´ and đ??ľ ⊆ đ??ş. This implies, đ??ľ ⊆ đ??´ ∊ đ??ş. Since B is strongly đ?‘”∗ - closed relative to A, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)) ⊆ đ??´ ∊ đ??ş. i.e., đ??´ ∊ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)) ⊆ đ??´ ∊ đ??ş. This implies, đ??´ ∊ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)) ⊆ đ??ş. Thus, đ??´ ∊ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)) âˆŞ (đ?‘‹ − (đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)))) ⊆ đ??ş âˆŞ (đ?‘‹ − đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ))). Also, đ??ľ ⊆ đ??´ which implies đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)) ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) Corollary 2.12: Let đ??´ be strongly đ?‘”∗ -closed and suppose that đ??š is closed then đ??´ ∊ đ??š is a strongly đ?‘”∗ -closed set. Proof: To show that đ??´ ∊ đ??š is strongly đ?‘”∗ -closed, we have to show đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´ ∊ đ??š)) ⊆ đ??ş whenever đ??´ ∊ đ??š ⊆ đ??ş and đ??ş is đ?‘”-open. Since đ??´ ∊ đ??š is closed in đ??´, we have đ??´ ∊ đ??š is strongly đ?‘”∗ -closed in đ??´. By Theorem 4.11, đ??´ ∊ đ??š is strongly đ?‘”∗ -closed in (đ?‘‹, đ?œ?1 , đ?œ?2 ), since đ??´ ∊ đ??š ⊆ đ??´ ⊆ (đ?‘‹, đ?œ?1 , đ?œ?2 ). Theorem 2.13: If đ??´ is strongly đ?‘”∗ -closed and đ??´ ⊆ đ??ľ ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)), then đ??ľ is strongly đ?‘”∗ -closed. Proof: Given that đ??ľ ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) then đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)) ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)), đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??ľ ⊆ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´. Since đ??´ ⊆ đ??ľ, and đ??´ is strongly đ?‘”∗ -closed, by Theorem 4.9, đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) − đ??´ contains no non-empty closed set and đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??ľ)) − đ??ľ contains no non-empty closed set. Again by Theorem 4.9, đ??ľ is a strongly đ?‘”∗ -closed set. Theorem 2.14: Let đ?‘‹ and đ?‘Œ are bitopological spaces and let đ??´ ⊆ đ?‘Œ ⊆ đ?‘‹ and suppose that đ??´ is strongly đ?‘”∗ - closed in đ?‘‹ then đ??´ is strongly đ?‘”∗ - closed relative to đ?‘Œ.
Cite this article as: J Logeshwari, J Manonmani, S Padmanaban, S Gowri Sankar. “STRONGLY g*-CLOSED SETS IN BITOPOLOGICAL SPACES.� International Conference on Systems, Science, Control, Communication, Engineering and Technology (2015): 167-169. Print.
International Conference on Systems, Science, Control, Communication, Engineering and Technology
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Proof: Given that đ??´ ⊆ đ?‘Œ ⊆ đ?‘‹ and đ??´ is strongly đ?‘”∗ -closed in đ?‘‹. To show that đ??´ is strongly đ?‘”∗ -closed relative to đ?‘Œ, let đ??´ ⊆ đ?‘Œ ∊ đ??ş, where đ??ş is đ?‘”-open in đ?‘‹. Since đ??´ is strongly đ?‘”∗ -closed in đ?‘‹, đ??´ ⊆ đ??ş implies đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊆ đ??ş. i.e., đ??´ ⊆ đ?‘Œâ‹‚đ??ş, where đ?‘Œ ∊ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) ⊆ đ?‘Œ ∊ đ??ş, where đ?‘Œ ∊ đ?œ?2 − đ?‘?đ?‘™(đ?œ?1 − đ?‘–đ?‘›đ?‘Ą(đ??´)) is the closure of interior of đ??´ in đ?‘Œ. Thus đ??´ is strongly đ?‘”∗ closed relative to đ?‘Œ. REFERENCES 1. 2. 3. 4. 5. 6.
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Cite this article as: J Logeshwari, J Manonmani, S Padmanaban, S Gowri Sankar. “STRONGLY g*-CLOSED SETS IN BITOPOLOGICAL SPACES.� International Conference on Systems, Science, Control, Communication, Engineering and Technology (2015): 167-169. Print.