General Topology (AN2)

Graduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN2
Semester	1
Course Title	General Topology
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Language of Instruction (lectures): Greek Language of Instruction (activities other than lectures): Greek and English Language of Examinations: Greek and English
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Using the Bloom Taxonomy. All the following sets are considered to be arbitrary subsets of an arbitrary Euclidean normed space of finite dimension. Remembering: Topological spaces, open and closed sets, interior and closure of sets. Continuous functions in topological spaces. Axioms of separation. Convergence in topological spaces. Metric spaces and metrizable spaces. Dimension of topological spaces. Dimension of metrizable space. Comprehension: Methods of generating topologies. Homeomorphisms. Frechet spaces. Operations on topological spaces. Functions spaces. Compact spaces, locally compact spaces, compactifications, countably compact spaces, pseudocompact spaces, sequentially compact spaces. Totally bounded and complete metric spaces. Paracompact spaces, countably paracompact spaces. Connected spaces, kinds of disconnectedness. Uniform spaces, totally bounded, complete and compact uniform spaces, proximity spaces. Applying: Thorough study of topological spaces. Thorough study of continuous functions in topological spaces. Evaluating: Teaching undergraduate courses.

General Competences

Production of free, analytic and inductive thinking. Required for the production of new ideas. Working independently. Team work. Decision making.

Syllabus

Topological spaces, methods of generating topologies, continuous mappings, axioms of separation, Frechet spaces, subspaces, Cartesian products, quotient spaces, function spaces, compact spaces, locally compact spaces, compactifications, countably compact spaces, pseudocompact spaces, sequentially compact spaces, totally bounded and complete metric spaces, paracompact spaces, countably paracompact spaces, connected spaces, kinds of disconnectedness, dimension of topological spaces and its basic properties, uniform spaces, totally bounded, complete and compact uniform spaces, proximity spaces.

Teaching and Learning Methods - Evaluation

Delivery	Lectures in class. Teaching is assisted by Learning Management System. Teaching is assisted by the use of online forums where students can participate in order to improve their problem solving skills, as well as their understanding of the theory they are taught. Teaching is assisted by the use of pre-recorded videos.
Use of Information and Communications Technology	Use of Learning Management System, combined with File Sharing Platform as well as Blog Management System for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, and communicating with students. Use of Appointment Scheduling System for organising appointments between students and the teacher. Use of Survey Web Application for submitting anonymous evaluations regarding the teacher. Use of Wiki Engine for publishing manuals regarding the regulations applied at the exams processes, the way teaching is organized, the grading methods, as well as the use of the online tools used within the course.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly presentations - oral exams, combined with weekly written assignments. In any case, all students can participate in written exams at the end of the semester. The aforementioned information along with all the required details are available through the course’s website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course’s website. Upon request, all the information is provided using email or social networks.

Attached Bibliography

1. Ryszard Engelking - General Topology
2. James Munkres - Topology
3. John Kelley - General Topology

1. Ryszard Engelking - General Topology.
2. James Munkres - Topology.
3. John Kelley - General Topology.