S.N

Content

Domain

Time (Hours)

1

 Statements only of (Axiom of choice, Zorn’s Lemma, Well ordering theorem and Continuum hypothesis)

Must know

10

 Definition and examples of topological spaces

 Neighbourhoods, Interior point and interior of a set

 Closed set as a complement of an open set

 Adherent point and limit point of a set,Closure of a set,Derived set

 Properties of closure operator, Boundary of a set, Dense subsets,Interior,Exterior and boundary operators

 Base and subbase for a topology

 Neighbourhood system of a point and its properties

 Basefor Neighbourhood system

 Relative(Induced) topology

 Alternative methods of defining a topolgy in terms of neighbourhood system and Kuratowski closure operator

 Comparison of topologies on a set,Intersection and union of topologies on a set

2


Must know

15

 Open and closed functions


 Connected and its characterization

 Connected subsets and their properties

 Continuity and connectedness



3

 Compact spaces and subsets

Must know

15

 Compactness in terms of finite intersection property

 Continuity and compact sets

 Basic properties of compactness

 Closedness of compactsubset and a continuous map from a compact space into a Hausdorff and its consequence

 Sequentially and countably compact sets

 Local compactness and one point compatification

4

 First countable, Second countable and separable spaces

Nice to know

12

 Hereditary and topological property

 Countability of a collection of disjoint open sets in separable and second countable spaces


 T0,T1,T2(Hausdorff) separation axioms, their characterization and basic properties
