About the course:This course begin with the assumption that the audience has attended the part-I. We cover all important advances topics such as Various notions of compactness in metric spaces, Ascolii’s theorem, paracompactness, compactifications, Urysohns Metrization theorem, Stone-Weiertrass theorem, totally disconnectedness, etc. Checking whether any of these topological properties in product invariant occupies a considerable amount of our time. A brief introduction to Topological Dimension Theory, discussion of function-space topologies and finally the Ordinal topology as a rich source of counter examples are the added features. The content of this course will be useful for any body who wishes to study deeper aspects of Analysis or topology or apply topological tools anywhere else in Mathematics or Physics, etc. It is certainly useful in the two course that I have offered on Algebraic Topology on NPTEL portal.PRE-REQUISITES: Attended the part-I of this course from NPTEL or acquired the basic knowledge the content offers from elsewhere.INTENDED AUDIENCE: Students from various streams which include some modern mathematics such as B. Sc., M. Sc., Students, Ph. D. , B. Tech. and M. Tech. who had not attended any topology courses seriously, before this, will be able to benefit form this course.
Overview
Syllabus
Week 1: Compactness and separation axioms.
Week 2:Paracompactness and partition of unity.
Week 3:Various notions of Compactness in metric metric spaces, Ascoli’s theorem.
Week 4:Productive properties.
Week 5:Productive properties continued.
Week 6:Urysohn’s Metrization theorem.
Week 7:Compactifications 1-pt compactification, Stone-Cech Compactification.
Week 8:Totally disconnectedness.
Week 9:A brief introduction to dimension theory.
Week 10:Function spaces, Compact open topology, exponential correspondence.
Week 11:Stone-Weierstass Theorem.
Week 12:Ordinal Topology- a source for many counter examples.
Taught by
Prof. Anant R. Shastri
