@MISC{Imura02finitetopological, author = {Hiroshi Imura and Masayoshi Eguchi}, title = {Finite Topological Spaces}, year = {2002} }

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Abstract

this paper. The following four propositions are true: number i such that 1 i and i < len f holds p i f p i+1 f : Let i, j be natural numbers. If i j and 1 i and j len f ; then p i f p j f : number i such that 1 i and i < len f holds p i f p i+1 f : Let i, j be natural numbers. Suppose i < j and 1 i and j len f and p j f p i f : Let k be a natural number. If i k and k j; then p j f = p k 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) 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) p 1 f = C ; (iii) for every natural number i such that i > 0 and i < len f holds p i+1 f =F (p i f ); (iv) F (p len f f ) = p len f f ; and (v) for all natural numbers i, j such that i > 0 and i < j and j len f holds p i f B and p i f p j f provided the following conditions are satisfied: B is finite, The proposition (4) has been removed. 1 c Association of Mizar Users C B ; and For every subset A of A such that A B holds A F (A) and F (A) B : We consider finite topology spaces as extensions of 1-sorted structure as systems h a carrier, a neighbour-map i, where the carrier is a set and the neighbour-map is a function from the carrier into 2 the carrier . One can verify that there exists a finite topology space which is non empty and strict. In the sequel F 1 is a non empty...