Paratopologies
by
Szymon Dolecki
Burgundy University

It had been gradually observed that various classical quotient maps (quotient, hereditarily quotient, bi-quotient, almost open) are the quotient maps in certain known reflective categories of convergences. However countably bi-quotient maps corresponded to no existent category. The class of paratopologies introduced by the author was the missing link.

Paratopologies turned out to be useful in category characterizations of various classes of maps and spaces, like countably perfect maps and strongly Fréchet spaces.

Several decades ago characterizations of sequential convergences induced by topologies were obtained by Kantorovich and his collaborators, Kisynski, and in non-Hausdorff case, by Kaminski. Greco and the present author obtained independently similar characterizations in non-Hausdorff case also for other subcategories of convergences. The class of Urysohn convergences remained a mysterious intermediate category between pseudotopologically and pretopologically induced sequential convergences.

Recently the author proved that Urysohn convergences are precisely paratopologically induced sequential convergences.

Date received: June 4, 2003