| An attempt is made to learn more about certain types of homogeneity which exist in some topological spaces and the relationships between them. Arguments constructed involve general topological considerations, transfinite induction, and a result from geometric topology. It is determined that if a T{dollar}sb1{dollar}-topological space possesses the property of being densely homogeneous, then all its components are densely homogeneous as well (i.e. dense homogeneity is inherited by components in densely homogeneous T{dollar}sb1{dollar} spaces). Using transfinite induction, an example from a class of spaces whose members are called Jones spaces is identified and shown not to be countable dense homogeneous. Jan van Mill (see (7)) showed the existence of a subset of the plane which is a Baire space and strongly locally homogeneous but not countable dense homogeneous. His is an existential construction. We improve upon van Mill's result by giving a constructive example which is a subset of the plane and has the same properties as van Mill's existential example. Let A = {dollar}{lcub}{dollar}X {dollar}varepsilon{dollar} R{dollar}sp2{dollar}: at least one coordinate of x is rational{dollar}{rcub}{dollar} and let B = {dollar}{lcub}{dollar}X {dollar}varepsilon{dollar} R{dollar}sp2{dollar}: either both coordinates of x are rational or both are irrational{dollar}{rcub}{dollar}. (B is the constructive example we alluded to.) We show the existence of a sequence of autohomeomorphisms of the plane that converges uniformly to an autohomeomorphism of the plane which witnesses the strong local homogeneity of A and B. The tame embedding theorem for planar arcs is employed in this effort. I am very grateful to Ben Fitzpatrick, Jr. for originating the problem that I discussed above.; Two problems this dissertation seeks to address but which remain unsolved are: (1) Is every open subset of a countable dense homogeneous metric space countable dense homogeneous? (2) Is there a connected metric space that is countable dense homogeneous but not strongly locally homogeneous?... |
