On the constructive notion of closure maps (Q2910990)

This paper is a contribution to the programme of constructive reverse mathematics, carried out in Bishop's constructive mathematics.NEWLINENEWLINEThe authors show that, under the assumption of certain additional intuitionistic principles, there exists no closure map such that both (1) all nontrivial open sets of \(\mathbb R\) are strictly included in their closure; and (2) the closure of a set \(A\) is the smallest functionally closed set of which \(A\) is a subset.NEWLINENEWLINEThroughout this paper, the authors introduce many additional principles that can be consistently added to Bishop's constructive mathematics, and they prove many positive and negative results on the notions of closed sets and closure maps. The negative results are obtained by way of Beth models.