CANKIRI KARATEKIN UNIVERSITY

Bologna Information System

Course Title | Code | Semester | Laboratory+Practice (Hour) | Pool | Type | ECTS |

INTRODUCTION TO TOPOLOGY | MAT305 | FALL | 4+0 | Fac./ Uni. | C | 8 |

Learning Outcomes | 1-To be able to explain basic properties of real numbers 2-To be able to do generalization of the concept of distance for real numbers to more general abstract metric spaces 3-To be able to determine pointwise and uniform convergence for sequences and series of real valued functions 4-To be able to explain concepts of connectedness and compactness in metric spaces |

Activity | Percentage (100) | Number | Time (Hours) | Total Workload (hours) |

Course Duration (Weeks x Course Hours) | 14 | 4 | 56 | |

Classroom study (Pre-study, practice) | 14 | 8 | 112 | |

Assignments | 10 | 2 | 8 | 16 |

Short-Term Exams (exam + preparation) | 10 | 2 | 8 | 16 |

Midterm exams (exam + preparation) | 30 | 1 | 12 | 12 |

Project | 0 | 0 | 0 | 0 |

Laboratory | 0 | 0 | 0 | 0 |

Final exam (exam + preparation) | 50 | 1 | 16 | 16 |

Other | 0 | 0 | 0 | 0 |

Total Workload (hours) | 228 | |||

Total Workload (hours) / 30 (s) | 7,6 ---- (8) | |||

ECTS Credit | 8 |

Week | Topics | Study Metarials |

1 | Upper and lower bounds of sets, Archimed property and dense sets in real numbers and related applications and properties, density properties of rational and irrational numbers in real numbers. | |

2 | The method of the nested intervals, definition of cluster point and their properties in real numbers and Balzano-Weierstrass Theorem. | |

3 | Real valued sequences and real valued Cauchy sequences and their definitions and related applications. | |

4 | Limit superior and limit inferior in real numbers, real valued functions sequences and series and pointwise convergence of real valued functions and series. | |

5 | Definition of uniform convergence of functions and function series, their properties, Dini Theorem. | |

6 | Derivative and uniform convergence in real valued functions sequences and series, uniform convergence and integral in real valued functions sequences and series. | |

7 | Weierstrass M-test, power series and their properties. | |

8 | Definition of metric space and examples, concept of neighbourhood, definitions of open and closed sets in the metric spaces and related other definition and applications. | |

9 | Sequences and Cauchy sequences in metric spaces and convergence of sequences in metric spaces. | |

10 | Complete metric spaces, related definitions and applications, non-density sets, first countable and second countable metric spaces and Baire Category Theorem. | |

11 | Completeness, open cover, compactness and compact metric spaces and their definitions and applications in metric spaces. | |

12 | Continuity, compactness, functions between metric spaces, continuity of them. | |

13 | Connected metric spaces and their properites. | |

14 | Relations between continuity and connectedness, Mean Value Theorem. |

Prerequisites | - |

Language of Instruction | Turkish |

Course Coordinator | Assoc. Prof. Dr. Faruk POLAT |

Instructors | - |

Assistants | - |

Resources | An introduction to real analysis, T. Terzioğlu, Matematik Vakfı yayınları, 2000 |

Supplementary Book | 1. The Elements of Real Analysis, R. G. Bartle, John Wiley and Sons, 1967 2. Real analysis with Real Applications, K. R. Davidson, A. P. Donsig, Prentice Hall, 2002 |

Document | - |

Goals | To introduce metric spaces and to state or introduce some well-known definitions and information that we know in Analysis I and II for metrices spaces. |

Content | Sets bounded from below and above in real numbers, the least upper bound property of real numbers and its consequences, Archimedean property of real numbers, density of rational and irrational numbers in real numbers, nested closed interval property in real numbers, definition of accumulation points and their propertes, Bolzano-Weierstrass Theorem, real valued sequences and their convergence, Cauchy sequences, Completeness of set of real numbers Limit superior and limit inferior in set of real numbers, sequences and series of real valued functions and their pointwise limit and uniform convergence, Dini Theorem, uniform convergence of sequences of functions and its integrals and derivatives, Weierstrass M-test for uniform convergence of series, Power series and their properties, definition of metric space and examples of metric spaces, neighborhoods in metric spaces, open and closed sets in metric spaces, sequences in metric spaces, Cauchy sequence in metric spaces, complete metric spaces and examples of complete metric spaces, nowhere dense sets, first and second countable metric spaces, Baire Category Theorem, completion of a metric space, open cover for a metric space, precompact metric space, compact metric spaces and their properties, continuous functions between metric spaces, connected metric spaces, Intermediate Value Theorem |

Program Learning Outcomes | Level of Contribution | |

1 | To have a grasp of theoretical and applied knowledge in main fields of mathematics | 5 |

2 | To have the ability of abstract thinking | 5 |

3 | To be able to use the gained mathematical knowledge in the process of identifying the problem, analyzing and determining the solution steps | 5 |

4 | To be able to relate the gained mathematical acquisitions with different disciplines and apply in real life | 2 |

5 | To have the qualification of studying independently in a problem or a project requiring mathematical knowledge | 3 |

6 | To be able to work compatibly and effectively in national and international groups and take responsibility | - |

7 | To be able to consider the knowledge gained from different fields of mathematics with a critical approach and have the ability to improve the knowledge | 4 |

8 | To be able to determine what sort of knowledge the problem met requires and guide the process of learning this knowledge | 4 |

9 | To adopt the necessity of learning constantly by observing the improvement of scientific accumulation over time | 2 |

10 | To be able to transfer thoughts on issues related to mathematics, proposals for solutions to the problems to the expert and non-expert shareholders written and verbally | 4 |

11 | To be able to produce projects and arrange activities with awareness of social responsibility | - |

12 | To be able to follow publications in mathematics and exchange information with colleagues by mastering a foreign language at least European Language Portfolio B1 General Level | - |

13 | To be able to make use of the necessary computer softwares (at least European Computer Driving Licence Advanced Level), information and communication technologies for mathematical problem solving, transfer of thoughts and results | - |

14 | To have the awareness of acting compatible with social, scientific, cultural and ethical values | - |

Çankırı Karatekin Üniversitesi Bilgi İşlem Daire Başkanlığı @
2017 - Webmaster