categorically-algebraic topology and its applications

this paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. it incorporates the most important settings of lattice-valued topology, including poslat topology of s.~e.~rodabaugh, $(l,m)$-fuzzy topology of t.~kubiak and a.~v{s}ostak, and $m$-fuzzy topology on $l$-fuzzy sets of c.~guido. moreover, its respective categories of topological structures are topological over their ground categories. the theory also extends the notion of topological system of s.~vickers (and its numerous many-valued modifications of j.~t.~denniston, a.~melton and s.~e.~rodabaugh), and shows that the categories of catalg topological structures are isomorphic to coreflective subcategories of the categories of catalg topological systems. this extension initiates a new approach to soft topology, induced by the concept of soft set of d.~molodtsov, and currently pursued by various researchers.