The space of digital Jordan curves in a Khalimsky digital plane is path-connected, and among the topologies studied, only the Khalimsky topology makes this space path-connected.
Barmak, Algebraic topology of finite topological spaces and applications, Lecture Notes in Mathematics, vol
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The Topological Complexity of Spaces of Digital Jordan Curves
The space of digital Jordan curves in a Khalimsky digital plane is path-connected, and among the topologies studied, only the Khalimsky topology makes this space path-connected.