Let $X =$ {0, 1, 2, ...} and $T$ = { $\emptyset$, $X$, {0}, {1}, {0,2}, {0,1,3}, {0,1,2,4}, {0,1,2,3,5}, ... } $\cup$ {{0,1,2}, {0,1,2,3}, {0,1,2,3,4}, ... }. It is easily verified that $T$ forms a topology on $X$. Burdick has shown (Amer. Math. Monthly January 2006 p. 83) that the singleton {0} generates infinitely many distinct sets in this space under the operations of closure, complement and union. Q: Is this the only countably infinite topological space (up to homeomorphism) in which a finite seed set generates an infinite family under those three operations?


The answer seems to be yes. Burdick's and my solutions to the above Monthly problem both used this space. Even before seeing his solution I was led to think it's the only one that works.


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(edited April 9 to correct the topology description)



Closure-complement-union: countable space, finite seed, infinite family, space unique?
If you replace each point in your space with 2 points (or more), and use the induced topology, then you still have the same generating property, but the resulting space has no isolated points, and hence is not homeomorphic to your space.


Perhaps you want to insist upon a weak separation axiom. 


