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Introduction to Topology

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About this book :-
Introduction to Topology written by NA

Book Detail :-
Title: Introduction to Topology
Pages: 118
Type: PDF
Language: English
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Book Contents :-
Introduction to Topology written by NA cover the following topics.
1. Topology
Metric Spaces, Open Sets (in a metric space), Closed Sets (in a metric space), Topological Spaces, Closed Sets (Revisited), Continuity, Homeomorphisms, Homeomorphism Examples, Theorems On Homeomorphism, Homeomorphisms Between Letters of Alphabet, Topological Invariants, Vertices, Holes, Classification of Letters, The curious case of the “Q”, Topological Invariants, Hausdorff Property, Compactness Property, Connectedness and Path Connectedness Propertie
2. Making New Spaces From Old
Cartesian Products of Spaces, The Product Topology, Properties of Product Spaces, Identification Spaces , Group Actions and Quotient Spaces
3. First Topological Invariants
Introduction, Compactness, Preliminary Ideas, The Notion of Compactness, Some Theorems on Compactness, Hausdorff Spaces, Spaces, Compactification, Motivation, One-Point Compactification, Theorems, Examples, Connectedness, Introduction, Connectedness, Path-Connectedness
4. Surfaces
Surfaces, The Projective Plane, RP2 as lines in R, 3 or a sphere with antipodal points identified, The Projective Plane as a Quotient Space of the Sphere, The Projective Plane as an identification space of a disc, Non-Orientability of the Projective Plane, Polygons, Bigons, Rectangles, Working with and simplifying polygons, Orientability, Definition, Applications To Common Surfaces, Conclusion, Euler Characteristic, Requirements, Computation, Usefulness, Use in identification polygons, Connected Sums, Definition, Well-definedness, Examples, RP, 2#T= RP, 2#RP, 2#RP, Associativity, Effect on Euler Characteristic , Classification Theorem, Equivalent definitions, Proof
5. Homotopy and the Fundamental Group
Homotopy of functions, The Fundamental Group, Free Groups, Graphic Representation of Free Group, Presentation Of A Group, The Fundamental Group, Homotopy Equivalence between Spaces, Homeomorphism vs. Homotopy Equivalence, Equivalence Relation, On the usefulness of Homotopy Equivalence, Simple-Connectedness and Contractible spaces, Retractions, Examples of Retractions, Computing the Fundamental Groups of Surfaces: The Seifert-Van Kampen Theorem, Examples, Covering Spaces, Lifting


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