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.

  • $\begingroup$ (For context, BD-N has a ton of equivalent formulations, and in particular it implies that sequentially continuous maps between metric spaces are continuous, that most variations of the fan theorem are the same, and it generally makes bounded subsets and sequences a lot nicer to work with) $\endgroup$
    – saolof
    Apr 8 at 12:42


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