Editing Mapping class group (section)

== Motivation ==
Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of [[homeomorphism]]s from the space into itself, that is, [[Continuous_function#Continuous_functions_between_topological_spaces|continuous]] maps with continuous [[Inverse function|inverses]]: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The [[open set]]s of this new function space will be made up of sets of functions that map [[Compact space|compact]] subsets ''K'' into open subsets ''U'' as ''K'' and ''U'' range throughout our original topological space, completed with their finite [[Intersection (set theory)|intersections]] (which must be open by definition of topology) and arbitrary [[Union (set theory)|unions]] (again which must be open). This gives a notion of continuity on the space of functions, so that we can consider continuous deformation of the homeomorphisms themselves: called [[Homotopy|homotopies]]. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the functional composition group structure already present on the space of homeomorphisms.