Malaysian Journal of Mathematical Sciences, May 2017, Vol. 11, No. 2
Hopf and Marcus-Wyse Topologies on \(\mathbb{Z}^2\)
Hegazy, K. and Mahdi, H.
Corresponding Email: hmahdi@iugaza.edu.ps
Received date: 1 November 2016
Accepted date: 12 April 2017
Abstract:
The two conditions \(1^2\) and \(2^2\) are so that any digital topology on \(\mathbb{Z}^2\) satisfies them is topologically connected whenever it is graphically connected. In this paper, we show that the two digital topologies on \(\mathbb{Z}^2\) satisfy \(1^2\) and \(2^2\) are precisely the Hopf and the Marcus-Wyse Topologies. We prove that the Hopf topology is the product of two Khalimsky topologies on \(\mathbb{Z}\). We also prove that the Hopf topology is homeomorphic to the cellular-complex topology on \(\mathbb{F}^2\) while the Marcus-Wyes topology is homeomorphic to \(\mathbb{Z}_{\left \{ 0,2 \right \}}^2\).
Keywords: Digital spaces, Alexandroff spaces, Hopf topology, Marcus- Wyes topology, cellular-complex topology
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