$\mathsf{GPK_\infty^+}$ is an alternative set theory in which we have comprehension for formulas which are positive in a certain sense; see [the SEP article](https://plato.stanford.edu/entries/settheory-alternative/#SystGPKiOlivEsse) for more detail *(or [this MO post](https://mathoverflow.net/a/379632/8133), which contains a nice description of one particular model of $\mathsf{GPK_\infty^+}$)*. Every model $X$ of $\mathsf{GPK_\infty^+}$ carries a natural topology $\sigma_X$, generated by a basis whose elements correspond to the elements of $X$ itself: for each $x\in X$, we make $\{y\in X: y\not\in x\}$ a basic open set. 

Now in $\mathsf{ZF}$-land, we have a **dichotomy** between sets and proper classes provided by size: roughly, a class is proper iff it surjects onto the ordinals. However, in a model $X$ of $\mathsf{GPK_\infty^+}$ we have a natural **trichotomy** amongst "$X$-classes:" actually an $X$-set, not an $X$-set but $\sigma_X$-closed, and not even $\sigma_X$-closed. I'm curious about whether $X$-classes in this second category are genuinely "more ($X$-)set-like;" this question is my attempt to make this vague idea precise.

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If $M,N$ are models of $\mathsf{GPK_\infty^+}$ with $M$ a submodel of $N$, say that $N$ is a **nice extension** of $M$ if

 - for each $m\in M$ we have $\{x\in M: M\models x\in m\}=\{y\in N: N\models y\in m\}$ (that is, $N$ is an **end extension** in the usual sense); and

 - $M$ is closed in $\sigma_N$.

Now say that a model $X$ of $\mathsf{GPK^+_\infty}$ is **rich** iff all $\sigma_X$-closed sets are $\sigma_X$-basic closed sets. I believe the usual construction of a model of $\mathsf{GPK^+_\infty}$ from a weakly compact cardinal produces a rich model, but not all models are rich (e.g. no countable model can be rich, and $\mathsf{GPK_\infty^+}$ is after all ["just"](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) a first-order theory). However, I don't immediately see an obstacle to every model being "potentially rich:"

> Is there an $M\models\mathsf{GPK_\infty^+}$ with no rich nice extension?

We could also look for a "single culprit" set:

> Is there an $M\models\mathsf{GPK_\infty^+}$ and a $\sigma_M$-closed $A\subset M$ such that no nice extension $N\supseteq M$ has an $n\in N$ with $A=\{x\in M: N\models x\in n\}$?

I don't have any ideas in either direction with respect to either question. On general grounds of orneriness I suspect that their answers are "yes" and "no" respectively, but this isn't really based on anything.

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An analogous question can be posed for pretty much any "topological" set theory. My specific focus on $\mathsf{GPK_\infty^+}$ is simply due to the fact that it is the topological set theory which seems most natural to me at the moment; I would be interested in answers for (non-contrived) other choices of topological set theories as well.