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Philadelphia University
Faculty of Science
Department of Basic Sciences and Mathematics
Second Semester, 2015/2016
Course Syllabus
Course code: 0250461
Course Title: Topology(1)
Course prerequisite (s) : Real Analysis(1)
Course Level: 4th
250311
Corequisite (s): Set Theory 250251
Lecture Time: 12:10-13:00 (Sun-Tue-Thu)
Credit hours: 3
Academic Staff Specifics
Name
Dr.
Mohammad
Al-khlyleh
Rank
Office Number
and Location
Office Hours
E-mail Address
10:10-11:10
Assistant
1019 S
(Sun-Tue-Thu)
professor
Faculty of Science
11:15-12:15
[email protected]
(Mon-Wed)
Prerequisite: Students are expected to have knowledge in propositional logic and quantification,
methods of proof, set operations and identities, relations and functions.
Course module description:
This is an introductory course in Topology. This course will provide a firm foundation in topology to
enable the student to continue more advanced study in this area. As several important areas of
mathematics, in particular modern analysis, depend upon or are clarified by certain topics in topology,
this course will present and emphasize those topics in order to aid the student in his future mathematical
studies. Finally, this course hopes to expose the students to both mathematical rigor and abstraction,
giving there an opportunity further to develop his mathematical maturity. Topics will include Topological
Spaces: Open sets, closed sets, closure, interior and boundary of a set, cluster points and the derived set,
isolated points. Relative topology and subspaces. Bases. Finite product of topological spaces. Continuous
functions, open functions, closed functions, homeomorphism, T0, T1 and T2 spaces, connected and
compact spaces.
Course module objectives:
This module aims to:






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Understand the concepts of topological spaces and apply them to different mathematical advanced
areas.
Learn and apply the concepts of topology on advanced courses.
Analyze and synthesize proofs to build proofs in a deductive reasoning.
Understanding the concepts of compactness especially for the real numbers and applying the idea
to different topological spaces.
Practicing proofs for many theorems on different ideas in topology to emphasis the right away in
building proofs.
Improving the student’s ability to think and write in a mature mathematical fashion and to a solid
understanding of the material most useful for advanced courses.
Course/ module components
Title: “An Introduction To General Topology”.
Author(s)/Editor(s): Long E. Paul.
Publisher: Jordan Book Center Company Limited, 1986.
ISBN-13: 978-0675092531.
ISBN-10: 0675092531.
Teaching methods:
Duration: 16 weeks, 48 hours in total.
Lectures: 48 hours, 3 per week + two exams (two hours).
Assignments: 3 quizzes.
Learning outcomes:

Knowledge and understanding
1. Understanding the basic topics of Topology, such as: the concepts; topology, topological spaces,
open sets, closed sets, closure, cluster points and compact.
2. Understanding the concepts of continuous functions and homoeomorphism.
3. Defining some examples of topological spaces, such as: discrete, indiscrete, usual, co-finite and cocountable topologies.

Cognitive skills (thinking and analysis).
1. Analyze and synthesize proofs to build proofs of topological theorems in a deductive reasoning.

Communication skills (personal and academic).
1. Display personal responsibility by working to multiple deadlines in complex activities.
2. Be able to work effectively alone or as a member of a small group working on some tasks.
3. Thinking and talking logically through the principle of proving a big amount of theorems.

Practical and subject specific skills (Transferable Skills).
1. Applying the concepts of topology to different mathematical advanced areas.
2. Practice operations on topological spaces and decide whether the result forms a topology.
Assessment instruments
Allocation of Marks
Assessment Instruments
Mark
First examination
20%
Second examination
20%
Final examination
40%
Quizzes, Home works
20%
Total
100%
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* Make-up exams will be offered for valid reasons only with consent of the Dean. Make-up exams may
be different from regular exams in content and format.
Course/module academic calendar
week
(1)
Basic and support material to be covered

Homework,
Reports and
their due dates
Topological Spaces:
 Defining a topology.
 Some examples.
(2)


Closed sets.
A closer look at the standard topology on R.

The Interior, Exterior and Boundary of a set.

Cluster points.

Topologies induced by functions.

Examples of topological spaces.

Relative topology and subspaces.
(3)
(4)
Quiz 1
(5)
(6)
First
exam
(7)
(8)

Bases, Subbases and Products:
 Bases.
Quiz 2
(9)


(10)
(11)
Second
exam.
(12)

Continuous functions:
 Defining a Continuous Function.
 Open functions, closed functions.


Finite products of topological spaces.
Subbases.
Homeomorphisms.
Separation and Countability Axioms:
 Separation axioms
(13)

Hausdorff Spaces
Quiz 3
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(14)

(15)
(16)
Final
Exam

The Second axiom of Countability and Separable
Spaces.
Compact Spaces:
 Compact Spaces and their properties.

Review.
Expected workload:
On average students need to spend 3 hours of study and preparation for each 50-minute lecture/tutorial.
Attendance policy:
Absence from lectures and/or tutorials shall not exceed 15%. Students who exceed the 15% limit
without a medical or emergency excuse acceptable to and approved by the Dean of the relevant
college/faculty shall not be allowed to take the final examination and shall receive a mark of zero for the
course. If the excuse is approved by the Dean, the student shall be considered to have withdrawn from
the course.
Module references
Students will be expected to give the same attention to these references as given to the Module textbooks.
Additional Books
1. Benjamin T. Sims, Fundamentals of Topology, 1976, Macmillan Publishing Co.
2. Seymour Lipschutz Kendall e. Atkinson, Theory and Problems of General Topology (Schaum’s
Outline Series), Schaum Publishing Co., ISBN: 0-471-02985-8.
3. Munkres, James R., Topology, 2nd Edition (2000), Upper Saddle River, New Jersey:
PrenticeHall, 2000, ISBN: 0-13-178449-8.
4. Willard,Stephen, GENERAL TOPOLOGY, London: Adelison-Wesley, 1970.
5. Armstrong, M. A, BASIC TOPOLOGY, New York: Springer, 2003.
Journals
--------------Mobile Sites
 http://www.goodreads.com/book/show/4495924-an-introduction-to-general-topology. Text book
website.