International Conference on Systems, Science, Control, Communication, Engineering and Technology
167
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.