On Boundary-Fixing Homotopies of Radal Homeomorphisms Defined on the Closed Planar Discs

In this work, homotopies of boundary-preserving homeomorphisms defined on the planar unit disc are studied. A set of radial homeomorphisms established in polar coordinates is considered. This set is subjected to rotating the circles, which have the same center, by an angle that depends in a continuous manner on the radius. However, these maps are homotopic to the identity map, the corresponding homotopies do not inevitably fix the boundary circle. For all parameter values, a point wise-fixed homotopy is defined between the homeomorphisms and the identity map. This establishes a concrete example and develops a technique for introducing homotopies restricted to a subspace. The outcomes support exploration of boundary-preserving transformations and planar homotopies in foundations of topology.