Finite Topological Spaces

(1) Let A be a set and f be a finite sequence of elements of 2A. Suppose that for every natural number i such that 1 i and i < len f holds πi f πi+1 f : Let i, j be natural numbers. If i j and 1 i and j len f ; then πi f π j f : (2) Let A be a set and f be a finite sequence of elements of 2A. Suppose that for every natural number i such that 1 i and i< len f holds πi f πi+1 f : Let i, j be natural numbers. Suppose i < j and 1 i and j len f and π j f πi f : Let k be a natural number. If i k and k j; then π j f = πk f : (3) Let F be a set. Suppose F is finite and F 6= / 0 and F is -linear. Then there exists a set m such that m 2 F and for every set C such that C 2 F holds C m: (5)1 Let f be a function. Suppose that for every natural number i holds f (i) f (i+1): Let i, j be natural numbers. If i j; then f (i) f ( j): The scheme MaxFinSeqEx deals with a non empty set A ; a subset B of A ; a subset C of A ; and a unary functor F yielding a subset of A ; and states that: There exists a finite sequence f of elements of 2 such that (i) len f > 0; (ii) π1 f = C ; (iii) for every natural number i such that i> 0 and i< len f holds πi+1 f =F (πi f ); (iv) F (πlen f f ) = πlen f f ; and (v) for all natural numbers i, j such that i > 0 and i < j and j len f holds πi f B and πi f π j f provided the following conditions are satisfied: B is finite,