What do you call a space whose only compact sets are finite? What do you call a topological space where a subset is compact iff it's finite? Is there a technical name?
For example, take the discrete topology, or the countable complement topology.
 A: As others have noted, the term that you want is anticompact. This is an example of the kind of property studied by Paul Bankston in The total negation of a topological property, Illinois J. of Math., Vol. 23, Nr. 2 (1979), 241-252. I quote:

Let $K$ be a topological class. The spectrum $\operatorname{Spec}(K)$ of $K$ is the class of cardinal numbers $\kappa$ such that any topology on a set of power $\kappa$ lies in $K$. For example, any topology on a finite set must be compact; and any infinite set supports noncompact topologies. Thus $\operatorname{Spec}(\{\text{compact spaces}\})=\omega$. Other spectra can be computed quite readily, such as $\operatorname{Spec}(\{\text{connected spaces}\})=2(=\{0,1\})$, and $\operatorname{Spec}(\{\text{perfect spaces}\})=1$.
Now let $K$ be a class of spaces and define $\operatorname{Anti}(K)$ to be the class of spaces $X$ such that whenever $Y\subset X$, $Y\in K$ iff $|Y|\in\operatorname{Spec}(K)$. Thus $X\in\operatorname{Anti}(K)$ iff the only subspaces of $X$ which are in $K$ are those which “have to be” on account of their cardinalities. Clearly $\operatorname{Anti}(K)$ is always hereditary.

