1. Preamble

1.1 Historical background

F. Hausdorff's Grundzüge der Mengenlehre (1914), considered to be the founding work of General Topology, although written in German, had a profound impact on topologists, who too often used it as their "bible", despite the more general book in French (admittedly more dissertative) by M. Fréchet (also a leading mathematician), entitled Les Espaces abstraits (1928). Fréchet (also a leading mathematician), entitled "Les Espaces abstraits" (1928), thus inhibiting the development of Pretopology until E. Čech's book in English (1966), obtained by translation and revision from the original Czech version (1959).

The foundations of pre-topology are intertwined with those of general topology, and date back to the early 20th century, with the work...