Question: 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)

Answer: 
The sum is zero for all $k<p^2-1$. 

Assume that $k$ is a multiple of $p-1$ and $k<p^2-1$. Divide all matrices into classes of the form $\{A,A+1,A+2,\dots,A+(p-1)\}$ . Summimg the $k$th powers over such a class yields
$$
 \sum\_{s=0}^k A^{k-s}\binom ks \sum\_{j=0}^{p-1} j^s
$$
where $0^0=1$. If $s=0$ or $s$ is not a multiple of $p-1$, the inner sum is 0 mod $p$. If $s$ is a multiple of $p-1$ and $s<k$, then the binomial coefficient is 0 mod $p$. (This is what breaks for $k=p^2-1$.) The only remaining term is the one $s=k$ and it does not depend on $A$. So the sum over a class is the same for all classes. The number of classes is a multiple of $p$, hence the result.
My answer was nonsense, sorry.