Can someone point me to any substitutes for the partition of unity in Bishop's constructive mathematics?


In particular, under what circumstances can we construct a partition of unity subordinate to some open cover of a compact set in Euclidean space?



Partitions of unity in constructive mathematics
In my PhD thesis [modern intuitionistic topology](https://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf) in chapter 3 there is a comprehensive treatment of partitions of unity within BISH.


In essence the main theorem is that per-enumerable open covers of a metric space admit a subordinate partition of unity. An open cover is per-enumerable iff it is an enumerable collection of open sets which themselves are an enumerable union of basic open metric balls (where the base point comes from a fixed enumerable dense set.) 


