Let us say that a topological space $X$ is a **Kreisel-Putnam** space when it satisfies the following property:

For all open sets $V_1, V_2$ and regular open set $W$ of $X$, if a point $x\in X$ has a neighborhood $N$ such that $N \cap W \subseteq V_1 \cup V_2$ then in fact it has a neighborhood $N'$ such that either $N' \cap W \subseteq V_1$ or $N' \cap W \subseteq V_2$. (Equivalently, $\operatorname{int}(V_1 \cup V_2 \cup (X\setminus W))$ is contained in the union of $\operatorname{int}(V_1 \cup (X\setminus W))$ and $\operatorname{int}(V_2 \cup (X\setminus W))$.)

(There may be clearer ways to phrase this. Perhaps passing to the closed complements is more palatable.)

The reason for this name and condition is that the above is equivalent to saying that the Heyting algebra of open sets of $X$ satisfies the Kreisel-Putnam axiom $(\neg u \Rightarrow (v_1\lor v_2)) \Rightarrow ((\neg u \Rightarrow v_1) \lor (\neg u \Rightarrow v_2))$ of interest in the study of intermediate logics. But of course, is this property has a different, more classical name, this is part of my question.

A *counterexample* is provided by $\mathbb{R}^2$: this does not satisfy the Kreisel-Putnam property, as shown by taking $V_1 = \{x_1 > 0\}$ and $V_2 = \{x_2 > 0\}$ and $W = V_1 \cup V_2$, which is indeed regular open, and the point $x = (0,0)$. (A slighly more complicated counterexample works for $\mathbb{R}$.)

Note the requirement that $W$ be regular open (or equivalently, be the pseudocomplement $\operatorname{int}(X\setminus U)$ of an open set $U$). If we drop this requirement, we get a (presumably!) stronger condition on $X$ which I might call a **Gödel-Dummett** space, because its Heyting algebra of open sets satisfies the axiom $(w \Rightarrow (v_1\lor v_2)) \Rightarrow ((w \Rightarrow v_1) \lor (w \Rightarrow v_2))$, which turns out to be equivalent to $(v_1 \Rightarrow v_2) \lor (v_2 \Rightarrow v_1)$, the Gödel-Dummett axiom.

As I find this property a little hard to visualize, I would like to ask:

Question:What are some interesting examples of Kreisel-Putnam spaces?

I don't have a precise definition of “interesting”, of course (I am trying to gain an intuitive grasp on the notion), but for example, discrete spaces (which are indeed Kreisel-Putnam) are definitely *not* interesting. Ideally, I would like something which is “somewhat like $\mathbb{R}^n$”, but some criteria which would help make a space interesting might be: being regular, connected (or at the very least, not extremally disconnected) and *not* satisfying the Gödel-Dummett condition.

It is probably worth pointing out that, as shown in this answer, the topos of simplicial sets satisfies the Kreisel-Putnam axiom. The corresponding condition for the topos of sheaves of sets on $X$ would be that every open set in $X$ satisfies the Kreisel-Putnam condition (maybe this is follows from merely $X$ satisfying it, this is one of the many things unclear to me).