P. 8


                    Notes             Explain the different kinds of topologies
                                      Solve the problems on intersection and union of topologies;
                                      Define open set and closed set;
                                      Describe the neighborhood of a point and solve related problems;

                                      Explain the dense set, separable space and related theorems and problems;
                                      Know the concept of limit point and derived set;
                                      Define interior and exterior of a set.


                                   Topology is that branch of mathematics which deals with the study of those properties of certain
                                   objects that remain invariant under certain kind of transformations as bending or stretching. In
                                   simple words, topology is the study of continuity and connectivity.
                                   Topology, like other branches of pure mathematics, is an axiomatic subject. In this, we use a set
                                   of axioms to prove propositions and theorems.

                                   This unit starts with the definition of a topology and moves on to the topics like stronger and
                                   weaker topologies, discrete and indiscrete topologies, cofinite topology, intersection and union
                                   of topologies, open set and closed set, neighborhood, dense set, etc.

                                   1.1 Topology and Different Kinds of Topologies

                                   1.1.1 Topology

                                   Definition 1: Let X be a non-empty set. A collection T of subsets of X is said to be a topology on
                                   X if
                                   (i)  X  T,   T

                                   (ii)  the intersection of any two sets in T belongs to T i.e. A T, B T  A  B T
                                   (iii)  the union of any (finite or infinite) no. of sets in T belongs to T.

                                       i.e. A  T      UA  T where  is an arbitrary set.
                                                            
                                       The pair (X, T) is called a Topological space.

                                          Example 1: Let X = {p, q, r, s, t, u} and T  = {X, , {p}, {r, s}, {p, r, s}, {q, r, s, t, u}}
                                   Then T  is a topology on X as it satisfies conditions (i), (ii) and (iii) of definition 1.

                                          Example 2: Let X = {a, b, c, d, e} and T  = {X, , {a}, {c, d}, {a, c, e}, {b, c, d}}
                                   Then T  is not a topology on X as the union of two members of T  does not belong to T .
                                        2                                              2                 2
                                                              {c, d}  {a, c, e} = {a, c, d, e}
                                   So, T  does not satisfy condition (iii) of definition 1.

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