In this paper we will introduce neutrosophic crisp Tri-topological spaces, and we will introduce
four new types of open and closed sets in neutrosophic crisp Tri-topological spaces. Then, the closure and
interior neutrosophic crisp set will be defined via this new concept of open and closed sets. Finally, we will
introduce the basic properties of these types of open and closed sets and the properties of new concept of
closure and the interior.
[1] R. K. Al-Hamido, Neutrosophic Crisp Bi-Topological Space, Communicated.
[2] K. Atanassov, intuitionistic fuzzy sets. in V.Sgurev, ed., Vii ITKRS Session, Sofia
(June 1983 central Sci. and Techn. Library, Bulg. Academy of Sciences (1984).
[3] K. Atanassov, intuitionistic fuzzy sets, fuzzy sets and systems 20, (1986) , 87-96.
[4] S. A. Alblowi, A. A. Salama and M. Eisa, New concepts of neutrosophic sets.
International Journal of Mathematics and Computer Applications Research (IJMCAR)
4, (2014), 59-66.
[5] S. A. Alblowi, A. A. Salama, and Mohmed Eisa. New concepts of neutrosophic sets.
International Journal of Mathematics and Computer Applications Research (IJMCAR)
3 , (2013), 95-102.
[6] I. Hanafy, A. A. Salama and K. Mahfouz, Correlation of neutrosophic data.
International Refereed Journal of Engineering and Science (IRJES) 1, (2012), 39-43.
[7] I. M. Hanafy, A. A. Salama and K. M. Mahfouz, Neutrosophic crisp events and its
probaTrility. International Journal of Mathematics and Computer Applications
Research (IJMCAR) 3, (2013), 171-178.
[8] F. Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference
on Neutrosophy, Neutrosophic Logic, Set, ProbaTrility, and Statistics, University of
New Mexico, NM 87301, USA(2002).
[9] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy,
Neutrosophic Set, Neutrosophic ProbaTrility. American Research Press, Rehoboth,
NM, (1999).
[10] F. Smarandache, An introduction to the Neutrosophy probaTrility applid in Quntum
Physics. International Conference on introducation Neutrosoph Physics, Neutrosophic
Logic, Set, ProbaTrility, and Statistics, University of New Mexico, Gallup, NM 87301,
USA2-4 December (2011).
[11] F. Smarandache, An introduction to the Neutrosophy probaTrility applid in Quntum
Physics. International Conference on introducation Neutrosoph Physics, Neutrosophic
Logic, Set, ProbaTrility, and Statistics, University of New Mexico, Gallup, NM 87301,
USA2-4 December (2011).
[12] A. A. Salama and S. A. Alblowi, Neutrosophic set and neutrosophic topological space.
ISORJ, Mathematics 3, (2012), 31-35.
[13] A. A. Salama and S. A. Alblowi, Generalized Neutrosophic Set and Generalized
Neutrousophic Topological Spaces. Journal computer Sci. Engineering 2, (2012), 29-
32.
[14] A. A. Salama and S. A. Alblowi, Intuitionistic Fuzzy Ideals Topological Spaces.
Advances in Fuzzy Mathematics 7, (2012), 51- 60.
[15] A. A. Salama, and H. Elagamy, Neutrosophic Filters. International Journal of
Computer Science Engineering and Information Technology Reseearch (IJCSEITR) 3,
(2013), 307-312.
[16] A. Salama, Neutrosophic Crisp Points and Neutrosophic Crisp Ideals. Neutrosophic
Sets and Systems 1, (2013), 50-54.
[17] A. Salama and F. Smarandache, Filters via Neutrosophic Crisp Sets. Neutrosophic
Sets and Systems 1, (2013), 34-38.
[18] A. A. Salama and S. A. Alblowi, Intuitionistic Fuzzy Ideals Topological Spaces.
Advances in Fuzzy Mathematics 7, (2012), 51- 60.
[19] D. Sarker, Fuzzy ideal theory, Fuzzy local function and generated fuzzy topology.
Fuzzy Sets and Systems 87, (1997), 117-123.
[20] A. Salama, F. Smarandache and V. Kroumov, Neutrosophic crisp Sets and
Neutrosophic crisp Topological Spaces, Neutrosophic Sets and Systems 2, (2014), 25-
30.
[21] A. Salama, Basic Structure of Some Classes of Neutrosophic Crisp Nearly Open Sets
and Possible Application to GIS Topology. Neutrosophic Sets and Systems 7, 2015,
18-22.
[22] F. G. Lupianez, On neutrosophic topology, Kybernetes 37 (2008), 797-800.
[23] F. G. Lupianez, Interval neutrosophic sets and topology, Kybernetes 38 (2009), 621-
624.
[23] F. G. Lupianez, On various neutrosophic topologies, Kybernetics 38 (2009), 1009-
1013.
[24] F. G. Lupianez, On neutrosophic sets topology, In F. Smarandache, & S. Pramanik
(Eds.), New trends in neutrosophic theory and applications. Pons Editions, Brussel,
2016, 305-313.
[25] L. A. Zadeh, Fuzzy Sets, Inform and Control 8, (1965), 338-353.
[1] R. K. Al-Hamido, Neutrosophic Crisp Bi-Topological Space, Communicated.
[2] K. Atanassov, intuitionistic fuzzy sets. in V.Sgurev, ed., Vii ITKRS Session, Sofia
(June 1983 central Sci. and Techn. Library, Bulg. Academy of Sciences (1984).
[3] K. Atanassov, intuitionistic fuzzy sets, fuzzy sets and systems 20, (1986) , 87-96.
[4] S. A. Alblowi, A. A. Salama and M. Eisa, New concepts of neutrosophic sets.
International Journal of Mathematics and Computer Applications Research (IJMCAR)
4, (2014), 59-66.
[5] S. A. Alblowi, A. A. Salama, and Mohmed Eisa. New concepts of neutrosophic sets.
International Journal of Mathematics and Computer Applications Research (IJMCAR)
3 , (2013), 95-102.
[6] I. Hanafy, A. A. Salama and K. Mahfouz, Correlation of neutrosophic data.
International Refereed Journal of Engineering and Science (IRJES) 1, (2012), 39-43.
[7] I. M. Hanafy, A. A. Salama and K. M. Mahfouz, Neutrosophic crisp events and its
probaTrility. International Journal of Mathematics and Computer Applications
Research (IJMCAR) 3, (2013), 171-178.
[8] F. Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference
on Neutrosophy, Neutrosophic Logic, Set, ProbaTrility, and Statistics, University of
New Mexico, NM 87301, USA(2002).
[9] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy,
Neutrosophic Set, Neutrosophic ProbaTrility. American Research Press, Rehoboth,
NM, (1999).
[10] F. Smarandache, An introduction to the Neutrosophy probaTrility applid in Quntum
Physics. International Conference on introducation Neutrosoph Physics, Neutrosophic
Logic, Set, ProbaTrility, and Statistics, University of New Mexico, Gallup, NM 87301,
USA2-4 December (2011).
[11] F. Smarandache, An introduction to the Neutrosophy probaTrility applid in Quntum
Physics. International Conference on introducation Neutrosoph Physics, Neutrosophic
Logic, Set, ProbaTrility, and Statistics, University of New Mexico, Gallup, NM 87301,
USA2-4 December (2011).
[12] A. A. Salama and S. A. Alblowi, Neutrosophic set and neutrosophic topological space.
ISORJ, Mathematics 3, (2012), 31-35.
[13] A. A. Salama and S. A. Alblowi, Generalized Neutrosophic Set and Generalized
Neutrousophic Topological Spaces. Journal computer Sci. Engineering 2, (2012), 29-
32.
[14] A. A. Salama and S. A. Alblowi, Intuitionistic Fuzzy Ideals Topological Spaces.
Advances in Fuzzy Mathematics 7, (2012), 51- 60.
[15] A. A. Salama, and H. Elagamy, Neutrosophic Filters. International Journal of
Computer Science Engineering and Information Technology Reseearch (IJCSEITR) 3,
(2013), 307-312.
[16] A. Salama, Neutrosophic Crisp Points and Neutrosophic Crisp Ideals. Neutrosophic
Sets and Systems 1, (2013), 50-54.
[17] A. Salama and F. Smarandache, Filters via Neutrosophic Crisp Sets. Neutrosophic
Sets and Systems 1, (2013), 34-38.
[18] A. A. Salama and S. A. Alblowi, Intuitionistic Fuzzy Ideals Topological Spaces.
Advances in Fuzzy Mathematics 7, (2012), 51- 60.
[19] D. Sarker, Fuzzy ideal theory, Fuzzy local function and generated fuzzy topology.
Fuzzy Sets and Systems 87, (1997), 117-123.
[20] A. Salama, F. Smarandache and V. Kroumov, Neutrosophic crisp Sets and
Neutrosophic crisp Topological Spaces, Neutrosophic Sets and Systems 2, (2014), 25-
30.
[21] A. Salama, Basic Structure of Some Classes of Neutrosophic Crisp Nearly Open Sets
and Possible Application to GIS Topology. Neutrosophic Sets and Systems 7, 2015,
18-22.
[22] F. G. Lupianez, On neutrosophic topology, Kybernetes 37 (2008), 797-800.
[23] F. G. Lupianez, Interval neutrosophic sets and topology, Kybernetes 38 (2009), 621-
624.
[23] F. G. Lupianez, On various neutrosophic topologies, Kybernetics 38 (2009), 1009-
1013.
[24] F. G. Lupianez, On neutrosophic sets topology, In F. Smarandache, & S. Pramanik
(Eds.), New trends in neutrosophic theory and applications. Pons Editions, Brussel,
2016, 305-313.
[25] L. A. Zadeh, Fuzzy Sets, Inform and Control 8, (1965), 338-353.