Course Name Code Semester T+U Hours Credit ECTS
Genelleştirilmiş Topoloji MAT 008 0 3 + 0 3 6
Precondition Courses <p>It is recommended that students to take Topology I and Topology II courses</p>
Recommended Optional Courses
Course Language Turkish
Course Level Doctorate Degree
Course Type Compulsory
Course Coordinator Prof.Dr. SOLEY ERSOY
Course Lecturers
Course Assistants
Course Category Field Proper Education
Course Objective

The aim of this course is to generalize some basic concepts of the topological space and to create infrastructure for the studies of doctoral students in the field of topology.

Course Content

Generalized topologies, regular open and regular closed sets, semi-open and semi-closed sets, pre-open and pre-closed sets, b-open and b-closed sets, generalized topologies and generalized neighborhood systems, complete generalized neighborhood systems, generalized continuity, semi-continuous functions, pre-continuous functions, irresolute functions, pre- irresolute functions, generalized open functions, generalized separation axioms, generalized weak separation axioms, generalized compactness, pre-compact, semi-compact, regular compactness, almost compactness, generalized normal spaces, generalized regular spaces, generalized connectedness.

# Course Learning Outcomes Teaching Methods Assessment Methods
1 He/she recalls the fundamental concepts of topology. Lecture, Question-Answer, Drilland Practice, Brain Storming, Problem Solving, Testing, Homework,
2 He/she generalizes the concepts of continuity, connectedness and compactness. Lecture, Question-Answer, Discussion, Group Study, Brain Storming, Problem Solving, Testing, Homework,
3 He/she analyzes and classifies the special cases of the generalized concepts. Lecture, Question-Answer, Discussion, Group Study, Brain Storming, Problem Solving, Testing, Homework,
4 He/she gives examples and contra- examples to each special cases Lecture, Question-Answer, Discussion, Group Study, Brain Storming, Self Study, Problem Solving, Testing, Homework,
5 He/She compares the concepts of each special case with each other. Lecture, Question-Answer, Discussion, Drilland Practice, Group Study, Brain Storming, Problem Solving, Testing, Homework,
6 He/She represents and proves related theorems. Lecture, Question-Answer, Discussion, Drilland Practice, Self Study, Problem Solving, Testing, Homework,
Week Course Topics Preliminary Preparation
1 Generalized Topologies
2 Regular Open And Regular Closed Sets, Semi-Open and Semi-Closed Sets, Pre-Open and Pre-Closed Sets, b-Open and b-Closed Sets
3 Generalized Topologies and Generalized Neighbourhood Systems
4 Complete Generalized Neighborhood Systems
5 Generalized Continuity
6 Semi-Continuous Functions, Pre-Continuous Functions, Irresolute Functions, Pre- Irresolute Functions
7 Generalized Open Functions
8 Generalized Separation Axioms
9 Generalized Weak Separation Axioms
10 Generalized Compactness
11 Pre-Compactness, Semi-Compactness, Regular Compactness
12 Generalized Normal Spaces
13 Generalized Regular Spaces
14 Generalized Connectedness
Resources
Course Notes <p>1. CSASZAR, A., On generalized neighbourhood systems. Acta Math. Hungar. 121 (2008), no. 4, 395400<br /> 2. LEVİNE, N., Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41.<br /> 3. MASHHOUR, A.S., ABD EL-MONSEF, M.E. and EL-DEEP, S.N., On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53.<br /> 4. NJASTAD, O., On some classes of nearly open sets, Pacific J. Math, 15 (1965), 961-970.<br /> 5. ABD EL-MONSEF, M.E., EL-DEEP, S.N. and MAHMOUD, R.A., open sets and continuous mappings, Bull Fac. Sci. Assiut Univ. A, A12, (1983), no. 1, 77-90.<br /> 6. JAMUNARANI, R. and JEYANTHI, P., Reg&uuml;ler sets in generalized topological spaces, Acta Math. Hungar., 135 (4) (2012), 342-349.<br /> 7. ANDRİJEVİC, D., On open sets, Mat. Vesnik, 48 (1996), 59-64.<br /> 8. CSASZAR, A., Generalized open sets, Acta Math. Hungar., 75 (1997), no. 1-2, 65-87.<br /> 9. CSASZAR, A., Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002), no. 4, 351-357.<br /> 10. CSASZAR, A., Generalized open sets in generalized topologies, Acta Math. Hungar., 106 (2005), 53-66.<br /> 11. EKİCİ, E., On weak structures due to Csaszar, Acta Math. Hungar., 134(4) (2012), 565-570.<br /> 12. CSASZAR, A., connected sets, Acta Math. Hungar., 101 (2003), 273-279.<br /> 13. SHEN R., A note on generalized connectedness, Acta Math. Hungar., 122 (2009), 231-235.<br /> 14. WU X. and ZHU P., A note on connectedness, Acta Math. Hungar., 139 (3) (2013), 252-254.<br /> 15. CSASZAR, A., Separation axioms for generalized topologies. Acta Math. Hungar. 104 (2004), no. 1-2, 63-69.</p>
Course Resources
Order Program Outcomes Level of Contribution
1 2 3 4 5
0 Develop strategic, political and practice plans and evaluate the results by taking into account the quality process in his/her area of expertise
1 At a master´s degree level, student reaches new knowledge via scientific researches, the use of knowledge of the same field as him/her or of different field from him/her, and the use of knowledge based on the competence in his/her field; s/he interprets the knowledge and prospects for the fields of application. X
2 Student completes the missing or limited knowledge by using the scientific methods. X
3 Student freely poses a problem of his/her field, develops a solution method, solves the problem, and evaluates the result. X
4 Student conveys, orally or in writing, his/her studies or the current developments in his/her field to the people in or out of his/her field. X
5 Student finds a solution to the unforeseen complex problems in his/her studies by developing new approaches. X
6 At a doctorate degree level, student prepares at least one scientific article of his/her field to be published in an international indexed journal, and s/he extends its popularity. X
7 Student analyzes the works that have been published before, approaches the same subjects with different proof methods, or determines the open problems about the current subject matters. X
8 Student looks for the scientists studying on the same field as him/her, and s/he gets in touch with them for a collaborative work. X
9 Student knows enough foreign language to do a collaborative work with the scientists studying on the same field as him/her abroad. X
10 Student follows the necessary technological developments in his/her field, and s/he uses them. X
11 Student looks out for the scientific and ethic values while gathering, interpreting and publishing data. X
Evaluation System
Semester Studies Contribution Rate
1. Ödev 100
Total 100
1. Yıl İçinin Başarıya 20
1. Final 80
Total 100
ECTS - Workload Activity Quantity Time (Hours) Total Workload (Hours)
Course Duration (Including the exam week: 16x Total course hours) 16 3 48
Hours for off-the-classroom study (Pre-study, practice) 16 3 48
Mid-terms 1 20 20
Assignment 1 10 10
Final examination 1 30 30
Total Workload 156
Total Workload / 25 (Hours) 6.24
dersAKTSKredisi 6