By Andrew O Lindstrum

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E. for any topological b) X space (Y, Y) , a ma p g : (X, X ) ~ (Y , Y) is continuous iff f or every i E I , g o I, : (X i, Xi) ~ (Y , Y ) is continuous. 1) Let (X, X ) and (Y,Y ) be to pological spaces. e. t. f. t. f. t. t he inclusion map i : A --+ X is denoted by X A and called t he relative topology; obviousl y, X A = {O n A : o E X} . (A , X A ) is called the subspace of (X , X) determined by A . CHA P T ER o. e. t . f. t. f. t. th e natural map w : X --+ X I R is denoted by X R , and (XI R , X R ) is called the quotient space of (X , X ) by R.

Let C be a topological construct , let X be a set and let ~ , TJ be C-structures on X . T he C- structure ~ is called finer t han 1] (or 1] coarser than 0 , denoted by ~ :::; 1], iff Ix : (X , ~) --t (X, 1]) is a C- morphism. The initial structure ~ on a set X with respect to (X , 1;, (Xi, ~i ) , I) in a topological construct C is the coarsest C- structure on X such that 1; is a C-morphism for each i E I . 5 Proposition. 3. 1) ~ is a C-structure on X such that all f i : (X,O --t (Xi, ~i) are C morphisms, Let T/ be a C st ruct ure on X such that all fi : (X ,1]) --t (Xi , ~i) are C-morphisms.

Pro~f. 8. 2. 1 D efinit io n . Let C be a topological construct and X a set. T he init ial C- st ru cture ';i (resp. 2. TEGOR ICA. 19 class I is called indiscrete (resp. discrete). 2 R emarks. 1) If ~i is the ind iscrete C-structure on X , t hen f (1", '1 ) --* (X, ~i) is a C-morphism for every obj ect (1",1]) and every map f 1" --* X . 2) If ~rI is the discret e C-structure, th en f : (X, ~d ) --* (Y, 1]) is a C-morphism for every C-object (1",1]) and every map f : X --* Y . 2 (3) t here is exactly one C- structure on X .