Every basic nearness (or
quasi-nearness) induces a Čech closure operator. There is a 1-1 correspondence
between the cluster generated Riesz nearnesses on a given T1 closure space and the
principal (or strict) T1 extensions of the space. (In particular linkage compact
extensions correspond to proximal nearnesses, F-linkage compact extensions
correspond to contigual nearnesses, while ordinary compact extensions correspond to
cluster generated weakly contigual nearnesses.
In this paper we discuss conditions under which a nearness-preserving map
between nearness spaces can be extended to a continuous map between the
corresponding principal extensions of the induced closure spaces. The concept of a
quasi-regular nearness space plays an important role in this connection. The general
results on extensions of nearness-preserving maps are used to obtain results on
extension of continuous maps into regular linkage compact and F-linkage compact
spaces.
