Proximity: a powerful tool in extension theory, function spaces, hyperspaces, boolean algebras and point-free geometry.

*(English)* Zbl 1192.54010
Mynard, Frédéric (ed.) et al., Beyond topology. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4279-9/pbk). Contemporary Mathematics 486, 89-114 (2009).

The article serves as a brief survey of the theory of proximity spaces and their usage in topology. Description and discussion of proximity axioms is provided, then inducing topologies, relations to uniformities and Smirnov compactifications follow, locally compact hulls are presented in more or less details. Finally, some applications to function spaces (mainly convergence), homeomorphism groups and to hyperspaces are given (connections to lattices and point-free geometry are briefly discussed, too).

##### MSC:

54E05 | Proximity structures and generalizations |

54B20 | Hyperspaces (general topology) |

54D35 | Extensions of topological spaces (compactifications, supercompactifications, completions, etc.) |

54-02 | Research monographs (general topology) |