Equivalent Topologies on the Contracting Boundary

The contracting boundary of a proper geodesic metric space generalizes the Gromov boundary of a hyperbolic space. It consists of contracting geodesics up to bounded Hausdorff distances. Another generalization of the Gromov boundary is the $\kappa$-Morse boundary with a sublinear function $\kappa$. The two generalizations model the Gromov boundary based on different characteristics of geodesics in Gromov hyperbolic spaces. It was suspected that the $\kappa$-Morse boundary contains the contracting boundary. We will prove this conjecture: when $\kappa =1$ is the constant function, the 1-Morse boundary and the contracting boundary are equivalent as topological spaces.