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let X={ a,b,c} with topology T= { empty set,{a},{a,b},{a,b,c,d} find all homomorphism f from x to x and justify that the list is exhaustive

let X={ a,b,c} with topology T= { empty set,{a},{a,b},{a,b,c,d} find all homomorphism f from x to x and justify that the list is exhaustive