Suppose N is the set of all positive integers and t consists of N, the empty set, and every set { n, n+1, ... } for n any positive integer. This is a topology and is called the “final segment topology.”

Is it the finite-closed topology? The finite-closed topology contains X (the "whole set"), the empty set, and all sets that have finite complements. These complements are the ONLY closed sets.

I think the answer is “yes.” If n = 1, then the “final segment” formed is N. If n = 4 (for example), then the complement is { 1, 2, 3 } which is a finite set, so all these “final segments” belong to this topology. There are other finite subsets, such as { 2, 4, 6 }, but they will be neither open nor closed, since they are not the complement of any “final segment” set.

Can anyone tell me if I'm right? Thanks in advance for any help.