Because I read the claim that intiutionistic logic is the logic of open subsets, let's describe a first-order semantics for a logic of open subsets, without equality and functions, but with many-sorted typing.
Let's assume we have a finite number of types α, β, ... for typing variables (so that a variable x of type α can be written as x:α) and a type *, which is the type of propositions. (Notation from The Seven Virtues of Simple Type Theory.) For each type α, a corresponding topological space X_α must be given. For each atomic proposition P, an open subset of the topological space X_* must be given. For each "atomic" predicate P(x:α, y:β), an open subset of the topological product space X_* x X_α x X_β must be given.
An open formula with free variables x:α and y:β will define an open subset of the topological product space X_* x X_α x X_β. If the classical interpretation of an operation like negation (¬P) or implication (P⇒Q) would lead to a non-open subset, it will be forced to an open subset by taking the (topological) interior of the non-open subset. The commonly used intuitionistic operations are the false proposition ⊥, the binary logical operations "and" ∧, "or" ∨, and "implication" ⇒, as well as the quantifier ∀ and ∃. For example ⊥ corresponds to the empty set (which is an open subset for any topological space), and the "and" P∧Q of two predicates P and Q is given by the intersection of the open subsets P and Q.
The definitions of P∨Q and ∃x:α.P are similarly unproblematic, but P⇒Q and ∀x:α.P require ensuring that the result is an open subset by taking the interior. The "implication" P⇒Q of two predicates P and Q is given by the interior of the union of the complement of P (in the corresponding product space) and the open subset Q. The universal quantification ∀x:α.P of a predicate P is given by the interior of the intersection over all x (in X_α) of the open subsets P|x.
Note that the semantics of constants was omitted, but the semantics of ∀x:α.P implicitly used a "very similar" concept. Defining the semantics of constants is easy (and doesn't require to take the interior of an intermediate result), so there seems to be no good reason to omit constants.
On the other hand, defining the semantics of equality would be quite annoying, because the formula x:α=y:α doesn't really define an open subset of the topological product space X_* x X_α x X_α. Taking the interior will definitively create confusion, because the formula x:α=x:α seems to correspond to the entire space X_* x X_α (which is an open subset). Note that for a normal predicate P(x:α, y:α), the formula P(x:α, x:α) canonically defines an open subset of X_* x X_α (and doesn't require to take the interior of an intermediate result), without any risk to create confusion. Hence there are good reasons to assume that omitting equality will simplify things enough to justify the occurred loss in expressivity.
Omitting the semantics of functions might be less clear cut for intuitionistic logic. Because nearly all first order formulations of intuitionistic logic include functions, I strongly suspect that they don't cause any harm. However, it still requires some work to figure out an appropriate semantics, and just because functions don't cause any harm for intuitionistic logic doesn't mean that functions won't cause any harm for partition logic either. So omitting functions should be OK, if equality is already omitted.
