A (non-empty/inhabited) subset $S$ of $\mathbb{N}$ is said to be pseudo-bounded if for every sequence $x_n$ in $S$ we have

$\lim_{n\to \infty} \frac{x_n}{n} = 0$

Clearly all bounded subsets are pseudo-bounded. The criteria BD is the classically true assertion that the converse holds and that pseudo-bounded subsets are bounded, and the weaker BD-$\mathbb{N}$ criterion requires the pseudo-bounded set to be countable for the converse to hold, which is a nontrivial extra condition in models where Kripkes schema (that subsets of countable sets are countable) does not hold. This is not assumed by bishop but BD-$\mathbb{N}$ holds very generally in both Russian and Brouwerian models, and for sheaves over $\mathbb{R}$. Lubarsky provides an example of a sheaf model where this does *not* hold.

So the question is then: are there good criteria for the topological models where BD-$\mathbb{N}$ and BD holds or do not hold? Can we relate it to standard topological properties like compactness/separability/countability of a space?

For more context on the axiom and many other related constructive axioms, there is an excellent overview article on constructive reverse mathematics by Hannes Diener that I was linked to in an answer to another question I had.