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

Generally speaking, topology is a closeness between points and sets, promixity is a closeness between sets and uniformity is a measure of closeness between sets made by a class of objects. Topology, proximity and uniformity are substantially distinct structures. Proximity, located between topology and uniformity, is a more powerful tool than topology having an intensive interaction with uniformity. Proximity is also an exhaustive machinery for the construction of T2-compactifications. Furthermore, proximity extends to complex frameworks such as function spaces, homeomorphism groups and hyperspaces. Then, proximity, in its dual formulation as strong inclusion, lends itself to a lattice theoretical approach. Tools, arguments, procedures and all techniques in the classic proximity theory are intensively employed in formalisations of pointfree geometries and in constructive topology. Proximity spaces have enough structure to feel comfortable within them!