Characterising near continuity constructively (Q2765576)

The authors provide some complements to their paper ``A constructive theory of point-set nearness'' [Theor. Comput. Sci. (to appear)], which in turn emerged from their fresh constructive approach to topology via apartness relations, initiated with ``Apartness as a framework for constructive topology'' [Ann. Pure Appl. Logic (to appear)]. They show that a mapping between arbitrary metric spaces respects nearness, between points and subsets, precisely when it preserves cluster points. Besides, any such mapping is proved to be strongly extensional and sequentially nondiscontinuous, but not sequentially continuous unless the domain is complete or one accepts Bishop's limited principle of omniscience. The present work is close in spirit to some work of \textit{Hajime Ishihara}'s: ``Continuity and nondiscontinuity in constructive mathematics'' [J. Symb. Log. 56, 1349-1354 (1991; Zbl 0745.03048)] and ``Continuity properties in constructive mathematics'' [J. Symb. Log. 57, 557-565 (1992; Zbl 0771.03018)].