VBS purvanchal University
M.Sc. 1st Year
Paper III: Topology
Unit I
Countable and uncountable sets, Infinite sets and the axiom of choice, Cardinal numbers
and its arithmetic, Schroeder-Bernstein theorem, Cantor’s Theorem and
continuum hypothesis, Zorn’s Lemma, Well ordering Theorem.
Unit II
Definition and examples of topological spaces, Closed sets. Closure, Dense subsets,
Neighbourhoods, Interior, exterior and boundary, Accumulation points and derived sets,
Bases and subbases, Subspaces and relative topology.
Unit III
Alternative methods of defining a topology in terms of Kuratoivski closure operator and neighbourhood systems, Continuous functions
homeomorphism
Unit IV
First & Second countable spaces, Lindeloff theorems. Separable
space second. Countibility and Separability.
Unit V
Separation axioms T0, T1, T2, T3, T4, their characterizations and basic properties . Uryson's lemma. Tietze extension theorem.
Unit VI
Compactness. Continous functions and compact sets. Basic properities of compactness. Compactness and finite intersection properity. Sequentially and countably compact sets, Local compactness and one point compactification. Stone-Cech compactification. Compactness in Metric spaces.
Unit VII
Connected spaces. Connectedness on the real time. Components, Locally connected spaces.
Unit VIII
Tychonoff product topology in terms of standard subbases
and its characteristics, Projection maps. Separation axioms and product spaces. Compactness, and product spaces (Tychonoff's Theorem).
Unit IX
Countability and product spaces.
Unit X
Embedding and Metrization. Embedding Lemma and Tychonoff embedding. The Urysohn's Metrization
Theorem.
Books Recommended :
1. James R. Munkers : "Topology - A First Course", Prentice-Hall of India Pvt. Ltd.,
New Delhi.
2. G. F. SImmons; "Introduction to Topology and Modern Analysis", Mc-Graw Hill.
3. K. D. Joshi; "Introduction to General Topology", Wiley Eastern.
4. J. Hocking and G. Young, Topology, Addison-Weslley, Reading, 1961
5. W. J. Pervin, Foundations of General Topology, Academic Press Inc. NY. 1964
6. K.K.Jha, Advanced General Topology, Nav Bharat Prakashan, Delhi
M.Sc. 1st Year
Paper III: Topology
Unit I
Countable and uncountable sets, Infinite sets and the axiom of choice, Cardinal numbers
and its arithmetic, Schroeder-Bernstein theorem, Cantor’s Theorem and
continuum hypothesis, Zorn’s Lemma, Well ordering Theorem.
Unit II
Definition and examples of topological spaces, Closed sets. Closure, Dense subsets,
Neighbourhoods, Interior, exterior and boundary, Accumulation points and derived sets,
Bases and subbases, Subspaces and relative topology.
Unit III
Alternative methods of defining a topology in terms of Kuratoivski closure operator and neighbourhood systems, Continuous functions
homeomorphism
Unit IV
First & Second countable spaces, Lindeloff theorems. Separable
space second. Countibility and Separability.
Unit V
Separation axioms T0, T1, T2, T3, T4, their characterizations and basic properties . Uryson's lemma. Tietze extension theorem.
Unit VI
Compactness. Continous functions and compact sets. Basic properities of compactness. Compactness and finite intersection properity. Sequentially and countably compact sets, Local compactness and one point compactification. Stone-Cech compactification. Compactness in Metric spaces.
Unit VII
Connected spaces. Connectedness on the real time. Components, Locally connected spaces.
Unit VIII
Tychonoff product topology in terms of standard subbases
and its characteristics, Projection maps. Separation axioms and product spaces. Compactness, and product spaces (Tychonoff's Theorem).
Unit IX
Countability and product spaces.
Unit X
Embedding and Metrization. Embedding Lemma and Tychonoff embedding. The Urysohn's Metrization
Theorem.
Books Recommended :
1. James R. Munkers : "Topology - A First Course", Prentice-Hall of India Pvt. Ltd.,
New Delhi.
2. G. F. SImmons; "Introduction to Topology and Modern Analysis", Mc-Graw Hill.
3. K. D. Joshi; "Introduction to General Topology", Wiley Eastern.
4. J. Hocking and G. Young, Topology, Addison-Weslley, Reading, 1961
5. W. J. Pervin, Foundations of General Topology, Academic Press Inc. NY. 1964
6. K.K.Jha, Advanced General Topology, Nav Bharat Prakashan, Delhi
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