Usual Topology || Discrete Topology || Indiscrete Topology || Topological space

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#M.Sc Maths #B.Sc NM #MCA # B.Tech. Mathematics.
Topology for M.Sc Mathematics
Topology Master Course
Topological Space
Theorems on Topology
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Definition and examples of topological spaces, closed sets and closure, dense subsets. Neighbourhoods interior, Exterior and boundary operations, Accumulation points and Derived sets. Bases and subbase. Subspaces and relative topology. Alternative method of defining a topology in terms of Kuratowski closure operator and neighbourhood systems. Continuous functions and homoemorphisms. Connected spaces. Connectedness on the real time. Components, Locally connected spaces.
Compactness, continuous functions and compact sets. Basic properties of compactness and finite intersection property. Sequentially and countably compact sets, Local compactness and one point compactification. Separation axioms T0,T1 and T2 spaces, Their characterization and basic properties, Convergence on T0 space.First and second countable spaces, Lindelof’s Theorems, Separable spaces and separability.
Unit-III
Regular and normal spaces, Urysohn’s Lemma and Tietze Extension Theorem, T3 and T4 spaces, Complete regularity and complete normality, T3 /½ and T5 spaces. Embedding and Metrization. Embedding Lemma and Tychonoff embedding Urysohn’s Metrization Theorem.
Unit-IV
Product topological spaces, Projection mapppings, Tychonoff product topology in terms of standard subbases and its characterization, Separation axioms and product spaces, Connectedness, locally connectedness and Compactness of product spaces. Product space as first axiom space.
Nets and filters. Topology and convergence of nets. Hausdorffness and nets. Compactness and nets. Filters and their convergence. Canonical way of converting nets to filters and vice-versa. ultra filters and compactness. Stone-Cech compactification.
Unit-V
Homotopy of paths, Fundamental group, Covering spaces, The fundamental group of the circle and fundamental theorem of algebra. Covering of a space, local finiteness, paracompact spaces, Michaell theorem on characterization of paracompactness in regular space, Paracompactness as normal, Nagata-Smirnov Metrization theorem
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Topology M.Sc Mathematics | Examples & Theorems | Abstract Algebra | Examples & Solution By Definition | Problems & Concepts by Karan Sir Mathematics will help Engineering and Basic Science students to understand following topic of Mathematics:
1. Vector Space
2. Subspaces
3. Homomorphism and Isomorphism
4. Linear Transformation
5. Basis of a vector space
6. Inner Product Space
Full course - Abstract Algebra
Advance Abstract Algebra
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This Concept is very important in Engineering & Basic Science Students. This video is very useful for B.Sc./B.Tech & M.Sc./M.Tech. students also preparing NET, GATE and IIT-JAM Aspirants.
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Dr. Karan Kamboj
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