Wednesday, June 13, 2018

general topology - Order type between two sets and bijection?

I want to show that {1,2}×Z+ and Z+×{1,2} have different order type



If we define f(i,j)=(j,i) for i in {1,2} and j in Z+



It seems like that this is bijective map between two sets.




However, to show that they are not order isomorphic, how shall I start to show that bijection does not preserve ordering?



It seems like that the way I defined the bijection is not the only way.



I am wondering if there exists any bijection between two sets and that bijection does not preserve order, can I conclude that they have different order type?

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