METRIC AND TOPOLOGICAL SPACES

Course Code:

MATH 439

Semester:

Fall

Course Type:

Core

P:

2

Lab:

2

Laboratuvar Saati:

0

Credits:

3

ECTS:

7

Prerequisite Courses:

REAL ANALYSIS I

Course Language:

English

Course Objectives:

To develop the necessary background for modern analysis courses to follow

Course Content:

Basic concepts about topological spaces and metric spaces. Complete metric spaces, Baire’s theorem, Contracting mapping theorem and its applications. Compact spaces, Arzela-Ascoli Theorem Seperability, second countability, Urysohn's lemma and the Tietze extension theorem, Connected spaces, Weierstrass approximation theoremA

Course Methodology:

1: Lecture, 2: Problem Solving

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Program Learning Outcomes	Teaching Methods	Assessment Methods
1) Learns basic concepts of topological spaces with emphasis on metric spaces		1,2	A
2) Learns Cauchy sequences and completeness		2,2	A
3) Learns the concept of compact space		1,2	A
4) Learns Baier’s category		1,2	A
5) Learns Ascoli-Arzela theorem, Weierstrass approximation		1,2	A
6) Acquires the skill of applying these concepts		1,2	A

Course Flow

Week	Topics	Study Materials
1	Basic concepts about metric spaces and examples
2	Open, closed sets, topology and convergence
3	Cauchy sequences and complete metric spaces, Baire's theorem
4	Continuity and uniformly continuity, spaces of continuous functions, Euclidean space
5	Contracting mapping theorem and its applications
6	The definition of topological spaces and some examples , elementary concepts, Open bases and open subbases
7	Compact spaces, Products of spaces,Tychonoff's theorem and locally compact spaces
8	Compactness for metric spaces
9	Arzela-Ascoli Theorem
10	Seperability, second countability
11	Hausdorff spaces, Completely regular spaces and normal spaces
12	Urysohn's lemma and the Tietze extension theorem
13	Connected spaces, The components of a space, Totally disconnected spaces, Locally connected spaces
14	The Weierstrass approximation theorem ,The Stone-Weierstrass theorems

Recommended Sources

Textbook

1. S. Kumaresan, Topology of Metric Spaces

2. George F. Simmons, Topology and Modern Analysis

3. W A Sutherland, Introduction to Metric and Topological Spaces

4. E T Copson, Metric Spaces

Additional Resources

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms	2	100
Quizzes	-	0
Assignments	-	0
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		60
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		40
Total		100

COURSE CATEGORY

Expertise/Field Courses

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)					X
2	Acquiring fundamental knowledge on fundamental research fields in mathematics					X
3	Ability form and interpret the relations between research topics in mathematics					X
4	Ability to define, formulate and solve mathematical problems					X
5	Consciousness of professional ethics and responsibilty			X
6	Ability to communicate actively			X
7	Ability of self-development in fields of interest					X
8	Ability to learn, choose and use necessary information technologies	X
9	Lifelong education			X

ECTS

ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION
Activities	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (14x Total course hours)	14	4	56
Hours for off-the-classroom study (Pre-study, practice)	14	5	70
Mid-terms (Including self study)	2	15	30
Quizzes	-
Assignments	-
Final examination (Including self study)	1	19	175
Total Work Load			175
Total Work Load / 25 (h)			7
ECTS Credit of the Course			7

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