Q:

Looking for non-trivial topologies satisfying certain conditions

I'm looking for topologies T on an infinite space X which divide the subsets of X into 2 non-empty collections: (1) sets which are both open and closed (clopen); (2) sets which are neither open nor closed.
The trivial topology is one example, with X and the null set being clopen and all other subsets neither open nor closed. But I haven't been able to find other examples. Nor have I been able to find an argument to show there can't be any others besides the trivial topology. 
Picture me as an old guy wandering around in Counterexamples in Topology and scratching his head, with his hair starting to fall out. Thanks for any help. 

A:

Combining Henning's and my comments and expanding them slightly:

The spaces whose topology consists of clopen sets only are precisely the disjoint unions of spaces with the trivial topology.

Let $X$ be a space whose topology $\tau$ consists of clopen sets only. Note that $\tau$ is stable under arbitrary unions and complements, hence it is also stable under arbitrary intersections.
Define a relation $\sim$ on $X$  as follows: $x \sim y$ if and only if for all clopen sets $U$ we have either $\{x,y\} \subset U$ or $\{x,y\} \cap U = \emptyset$.
This is obviously a reflexive and symmetric relation. It is also transitive:
Suppose $x \sim y$ and $y \sim z$. Consider an arbitrary clopen set $U$. By definition of $x \sim y$ there are only two possibilities:

If $\{x,y\} \cap U = \emptyset$ then we must have $\{y,z\} \cap U = \emptyset$ since $y \sim z$, hence $\{x,z\} \cap U = \emptyset$.
If $\{x,y\} \subset U$ then we must also have $\{y,z\} \subset U$ since $y \sim z$ and hence $\{x,z\} \subset U$.

Thus $x \sim z$ and the relation is transitive.
Write $[x]$ for the $\sim$-equivalence class of $x$. We have just seen that for each clopen $U$ we either have $[x] \subset U$ or $[x] \cap U = \emptyset$. Thus each clopen set $U$ is a union of equivalence classes. If $x \not \sim y$ then there exists a clopen set $V$ such that $[x] \subset V$ and $[y] \cap V = \emptyset$. Thus $[x]$ is the intersection of the clopen sets containing it and hence it is clopen as well. Since each clopen set either intersects $[x]$ trivially or must contain it, the relative topology on $[x]$ is the trivial topology. Since $X$ is partitioned into equivalence classes, it follows that $X$ is a disjoint union of spaces with the trivial topology.
Conversely, if $X$ is a disjoint union of spaces with the trivial topology then it has the desired property.

Finally, as Henning observed, every equivalence relation on $X$ endows $X$ with a topology consisting of clopen sets only: the topology generated by the equivalence classes. The identity relation gives the discrete topology and the trivial relation $X \times X$ gives the trivial topology. Every other equivalence relation gives a topology as you're asking for.

Added: From your comment I learned that this is called the partition topology by Steen and Seebach in Counterexamples in Topology and appears as Example 5 on page 43.

