In the category of topological spaces with homeomorphisms, object groups are exactly homeomorphism groups. In these cases the flow forms a one parameter subgroup of homeomorphisms and diffeomorphisms, respectively. Mapping tori of homeomorphisms of surfaces are of particular importance in 3-manifold topology. In particular the group of isometries of formula 8 acts by homeomorphisms on formula 108. A topological property is a property of spaces that is invariant under homeomorphisms. In that case one of the homeomorphisms can be chosen to be a diffeomorphism. Given by piecewise projective homeomorphisms of the line, the group is remarkably simple to understand. The property of being locally Euclidean is preserved by local homeomorphisms. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions. Since coverings are local homeomorphisms, a covering of a topological "n"-manifold is an "n"-manifold. The deformations that are considered in topology are homeomorphisms and homotopies. The homeomorphisms may be iterated, resulting in an iterated function system. A significant contribution in topology is his obstruction theory for the smoothing of homeomorphisms. We can also define a topology on this new space of homeomorphisms. These homeomorphisms are the charts of the manifold.