ON MON01DAL CLOSED TOPOLOGICAL CATEGORIES I Manfred B. Wischnewsky_ The aim of this paper initiated by L.D. Nel's talk at the Con- ference on Categorical Topology at Mannheim (1975) is to give two dif- ferent characterizations of monoidal closed topological categories. Both of these characterizations include as special instance the Herrlich-Nel results on cartesian closed (relative) topological cate- gories*over the base category Sets ([11], ~3~ ). The main tools in this paper are a generalization of the Special Adjoint Functor Theorem - the Relative Special AdJoint Functor Theorem - and the Dubuc-Tholen theory of Adjoint Triangles. As corollaries we obtain a characteri- zation of monoidal closed (relative) Top-categories over wellbounded (= locally bounded) categories and over cocomplete, wellpowered and cowellpowered monoidal closed categories. It is assumed that the reader is familiar with the notation and the content of ~12]. All other notions which are used in this paper are briefly recalled. w I Factorizations and Generators, Review. Let us first recall some of the basic notions and propositions on factorizations and generators (see e.g. ~9], ~6], ~8a]). Let A be a cate- gory. For two morphisms e : A-~B and m : C-~D we write e~m if every commutative diagram ge = mf can be made commutative by a unique morphism w : B---+C. If PcMor(A) then let P~:= [e: e~m for all mEP]and P+:=[m: eCru for all e~P).Let E,McMorA.Then (E,M) is called a prefactori- nation in A if E~= M and M@= E. A prefactorization (E,M) is called a fac- torization if every morphism f in A is of the form f = me with e ~E and m ~M. A factorization (E,M) is called proper if every e ~ E is an *) Relative topological category ~ E- reflective subcategory of an (absolute) topological category.