  In continuous first-order logic, the union of definable sets is definable but
generally the intersection is not. This means that in any continuous theory,
the collection of $\varnothing$-definable sets in one variable forms a
join-semilattice under inclusion that may fail to be a lattice. We investigate
the question of which semilattices arise as the collection of definable sets in
a continuous theory. We show that for any non-trivial finite semilattice $L$
(or, equivalently, any finite lattice $L$), there is a superstable theory $T$
whose semilattice of definable sets is $L$. We then extend this construction to
some infinite semilattices. In particular, we show that the following
semilattices arise in continuous theories: $\alpha+1$ and $(\alpha+1)^\ast$ for
any ordinal $\alpha$, a semilattice containing an exact pair above $\omega$,
and the lattice of filters in $L$ for any countable meet-semilattice $L$. By
previous work of the author, this establishes that these semilattices arise in
stable theories. The first two are done in languages of cardinality $\aleph_0 +
|\alpha|$, and the latter two are done in countable languages.
