Page 164 - DMTH503_TOPOLOGY
P. 164

Topology




                    Notes          Thus, G and H are open sets such that
                                   x  G but y  G
                                   and y  H but x  H
                                   Hence, the space (X, T) is a T -space.
                                                         1
                                   Conversely, let us consider the cofinite topology T on an infinite set X.
                                   Let x be an arbitrary point of X.
                                   by definition of T,
                                   X – {x} is open, for {x} is finite set and so {x} is T-closed.

                                   Thus, every singleton subset of X is closed.
                                   It follows that the space (X, T) is a T -space. Now we shall show that the space (X, T) is not a
                                                                1
                                   T -space.
                                    2
                                   For this topology, no two open subsets of X can be disjoint.
                                   Let if possible G and H be two open disjoint subsets of X, then
                                       G   H = 
                                      (G   H)’ = ’

                                      G’   H’ = X   (by De-Morgan’s law)
                                   Here G’   H’ being the union of two finite sets is finite, where as X is infinite.
                                   Hence for this topology no two open sets can  be disjoint i.e. no two distinct  points can  be
                                   separated by open sets.
                                   Hence, (X, T) is not T -space.
                                                   2
                                   Theorem 5: Every subspace of a T -space is a T -space
                                                             2         2
                                                   or

                                   Prove that every subspace of a Hausdorff space is also Hausdorff.
                                   Proof: Let (X, T) be a Hausdorff space and (Y, T ) be a subspace of it.
                                                                         y
                                   Let x and y be any two distinct points of Y.
                                   Then x and y are distinct points of X.

                                   But (X, T) is a Hausdorff space,  T-open nhds. G and H of x and y respectively such that
                                                                     G   H = 
                                   Consequently, Y   G and Y   H are T -open nhds of x and y respectively.
                                                                 y
                                   Also x  G, x  Y  x  Y   G

                                   and y  H, y  y  y  Y   H
                                   and since G   H = , we have
                                   (Y   G)   (Y   H) = Y   (G   H) = Y    = 
                                   This shows that (Y, T ) is also a T -space. Hence, every subspace of a Hausdorff space is also a
                                                    y         2
                                   Hausdorff space.






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