Hopf and Marcus-Wyse Topologies on ℤ2

The two conditions 12 and 22 are so that any digital topology on ℤ2 satisfies them is topologically connected whenever it is graphically connected. in this paper,we show that the two digital topologies on ℤ2 satisfy 12 and 22 are precisely the hopf and the marcus-wyse topologies. we prove that the hopf topology is the product of two khalimsky topologies on ℤ. we also prove that the hopf topology is homeomorphic to the cellular-complex topology on f2 while the marcus-wyes topology is homeomorphic to f(0;2) 2.