Examples of Kreisel-Putnam topological spaces

Question

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).

Answer 1

Some observations (hopefully correct):

1. It is enough to consider $x ∈ ∂W$ and $V_1, V_2 ⊆ W$. So for fixed $x, W$ we want that $\mathcal{N}_{x, W} := \{N ∩ W: N$ open neighborhood of $x\}$ is a prime filter (in the distributive lattice of open subsets of $W$).
2. Extremally disconnected spaces are exactly those where every regular open set is clopen, so such spaces are Kreisel–Putnam simply because of lack of suitable regular open sets. (That is probably why you are not interested in such examples).
3. The space $\{0, 1, 2\}$ with the topology generated by $\{1\}, \{2\}$ is a simple example of a connected Kreisel–Putnam space that is neither Gödel–Dummet nor extremally disconnected.
4. The space $\{0, 1, 2, 3\}$ with the topology generated by $\{1, 3\}, \{2, 3\}$ is a simple example of a connected extremally disconnected (and so Kreisel–Putnam) space that is not Gödel–Dummet.
5. A $T_1$ connected Gödel–Dummet space has no cut-points, i.e. $X \setminus \{x\}$ is connected for every $x$. Proof: If $X \setminus \{x\} = V_1 ∪ V_2$ for some disjoint nonempty open sets, we put $W := V_1 ∪ V_2$ and have a contradiction since $x ∈ \overline{V_1} ∩ \overline{V_2}$.
6. No Hausdorff Kreisel–Putnam space contains a nontrivial convergent sequence, so it is quite far from being “somewhat like $ℝ^n$”. Proof: Let $x_n \to x$ be a nontrivial convergent sequence. There are pairwise disjoint respective open neighborhoods $U_n$ not nontaining $x$. Now $W := \operatorname{int} \overline{⋃_{n ∈ ω} U_{2n}}$ is a regular open set with $x ∈ ∂W$ and $V_1 := W \setminus \{x_{4n}: n ∈ ω\}$ and $V_2 := W \setminus \{x_{4n + 2}: n ∈ ω\}$ are its open subsets such that $W = V_1 ∪ V_2$, yet $x$ is in the closure of their complements.

It seems that a maximal connected expansion of the real line is a Hausdorff connected Kreisel–Putnam space that is not Gödel–Dummet. So let $X$ be such expansion, which exists by [1], [2].

7. For every $x$, $\mathcal{N}_{x, (x, ∞)}$ is a maximal filter of open subsets of $(x, ∞)$. Hence, for every open $W ⊆ (x, ∞)$ such that $x ∈ ∂W$ we have $V ∈ \mathcal{N}_{x, W}$ iff $x ∈ \overline{V}$. The same holds for $(-∞, x)$. So $X$ is even Gödel–Dummet at pairs $x ∈ ∂W$ such that $W$ is only on one side.
8. On the other hand, $X$ is not really Gödel–Dummet by 5. since every point of $X$ is a cut-point: $X = (-∞, x) ∪ \{x\} ∪ (x, ∞)$.
9. By an almost clopen set at $x$ we mean an open set $U ≠ ∅$ such that $U ∪ \{x\}$ is closed. Note that in our $X$ there are only few almost clopen sets. If $U ⊆ X$ is almost clopen at $x$, then $U ∩ (x, ∞)$ is clopen in $(x, ∞)$. But $(x, ∞)$ is connected in $X$ since every connected expansion of $ℝ$ has the same connected sets as $ℝ$ [3]. So $U ∩ (x, ∞) ∈ \{∅, (x, ∞)\}$. The same holds for $U ∩ (-∞, x)$.
10. To show that $X$ is Kreisel–Putnam let $x ∈ ∂W$ for some regular open set $W ⊆ X$. Since $X$ is nearly maximal connected [4], there is $U$ a neighborhood of $x$ and an almost clopen set $V$ at $x$ such that $U ∩ V ∩ W = ∅$. Since by 9. we have $V = (-∞, x)$ or $(x, ∞)$, $W$ is essentially only on one side of $x$, and so we may apply 7.

Therefore, being Kreisel–Putnam look like quite exotic property, which may be connected with the notions of maximal connectedness and near maximal connectedness. Existence of a regular maximal connected space is an open problem. Recently [5], it was shown that there is no regular maximal connected expansion of $ℝ$.

[1] Simon, Petr, An example of maximal connected Hausdorff space, Fundam. Math. 100, 157-163 (1978). ZBL0435.54017.

[2] Guthrie, J. A.; Stone, H. E.; Wage, M. L., Maximal connected expansions of the reals, Proc. Am. Math. Soc. 69, 159-165 (1978). ZBL0396.54001.

[3] Hildebrand, S. K., A connected topology for the unit interval, Fundam. Math. 61, 133-140 (1967). ZBL0185.26101.

[4] Clark, Bradd; Schneider, Victor, A characterization of maximal connected spaces and maximal arcwise connected spaces, Proc. Am. Math. Soc. 104, No. 4, 1256-1260 (1988). ZBL0692.54010.

[5] Kalapodi, A.; Tzannes, V., The non-existence of a regular maximal connected expansion of the reals, Topology Appl. 232, 34-38 (2017). ZBL1377.54026.