Jordan Curve Theorems with Respect to Certain Pretopologies on Z~2

摘要

We discuss four quotient pretopologies of a certain basic topology on Z~2. Three of them are even topologies and include the well-known Khalimsky and Marcus-Wyse topologies. Some known Jordan curves in the basic topology are used to prove Jordan curve theorems that identify Jordan curves among simple closed ones in each of the four quotient pretopologies.