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.
At first, I thought the answer is “yes.” I reasoned like this: 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.
After more thought, I changed my mind. Here's my new reason: consider a singleton set in N such as { 3 }. It is finite, and its complement in N is infinite, so it is a closed set in the "finite closed" topology. It's complement is { 1 , 2, 4 , 5 ... } which is open in the "finite closed" topology, but it is not a "final segment," so it's not open in the "final segment" topology. If I'm right, this means the two topologies have different set of open sets, and so are different.
Am I right about that? Thanks. R
