Co-Hopf Space Structure on Closure Spaces

By constructing Hopf costructures on closure spaces via homotopy, we give the concepts of cospace (CH-cospace) and cogroup (CH-cogroup). We then prove that retract deformation a CH-cospace are also CH-cospace. construct costructure set with help quotient operator. show space same homotopy type as CH-cogroup is itself CH-cogroup. existence covariant functor between category pointed ($\mathcal{CHC}$) groups homomorphisms.