Connectedness Conditions Using Polygonal Arcs

Transcription

1 JOURNAL OF FORMALIZED MATHEMATICS Volume 4, Released 199, Published 003 Inst. of Computer Science, Univ. of Białystok Connectedness Conditions Using Polygonal Arcs Yatsuka Nakamura Shinshu University Nagano Jarosław Kotowicz Warsaw University Białystok Summary. A concept of special polygonal arc joining two different points is defined. Any two points in a ball can be connected by this kind of arc, and that is also true for any region inet. MML Identifier: TOPREAL4. WWW: The articles [1], [14], [], [8], [1], [15], [4], [3], [13], [11], [10], [9], [5], [6], and [7] provide the notation and terminology for this paper. For simplicity, we use the following convention: P, P 1, P, R denote subsets ofet, W denotes a non empty subset ofet, p, p 1, p, q denote points ofet, f, h denote finite sequences of elements ofet, r denotes a real number, u denotes a point ofe, and n, i denote natural numbers. The following proposition is true (1) Let D be a non empty set, f be a finite sequence of elements of D, and p be an element of D. Then ( f p ) len f+1 = p. Let us consider P, p, q. We say that P is a special polygonal arc joining p and q if and only if: (Def. 1) There exists f such that f is a special sequence and P = L( f) and p = f 1 and q = f len f. Let us consider P. We say that P is special polygon if and only if the condition (Def. ) is satisfied. (Def. ) There exist p 1, p, P 1, P such that p 1 p and p 1 P and p P and P 1 is a special polygonal arc joining p 1 and p and P is a special polygonal arc joining p 1 and p and P 1 P = {p 1, p } and P = P 1 P. We introduce P is a special polygon as a synonym of P is special polygon. Let P be a subset ofet. We say that P is region if and only if: (Def. 3) P is open and connected. We introduce P is a region as a synonym of P is region. Next we state a number of propositions: () If P is a special polygonal arc joining p and q, then P is a special polygonal arc. (3) If W is a special polygonal arc joining p and q, then W is an arc from p to q. (4) If W is a special polygonal arc joining p and q, then p W and q W. (5) If P is a special polygonal arc joining p and q, then p q. 1 c Association of Mizar Users

2 CONNECTEDNESS CONDITIONS USING POLYGONAL ARCS (6) If W is a special polygon, then W is a simple closed curve. (7) Suppose p 1 = q 1 and p q and p Ball(u,r) and q Ball(u,r) and f = p,[p 1, p +q ], q. Then f is a special sequence and f 1 = p and f len f = q and L( f) is a special polygonal arc joining p and q and L( f) Ball(u,r). (8) Suppose p 1 q 1 and p = q and p Ball(u,r) and q Ball(u,r) and f = p,[ p 1+q 1, p ], q. Then f is a special sequence and f 1 = p and f len f = q and L( f) is a special polygonal arc joining p and q and L( f) Ball(u,r). (9) Suppose p 1 q 1 and p q and p Ball(u,r) and q Ball(u,r) and [p 1,q ] Ball(u,r) and f = p,[p 1,q ],q. Then f is a special sequence and f 1 = p and f len f = q and L( f) is a special polygonal arc joining p and q and L( f) Ball(u,r). (10) Suppose p 1 q 1 and p q and p Ball(u,r) and q Ball(u,r) and [q 1, p ] Ball(u,r) and f = p,[q 1, p ],q. Then f is a special sequence and f 1 = p and f len f = q and L( f) is a special polygonal arc joining p and q and L( f) Ball(u,r). (11) If p q and p Ball(u,r) and q Ball(u,r), then there exists P such that P is a special polygonal arc joining p and q and P Ball(u, r). (1) Suppose p f 1 and ( f 1 ) = p and f is a special sequence and p L( f,1) and h = f 1, [ ( f 1) 1 +p 1,( f 1 ) ], p. Then (vi) L(h) = L( f 1) L( f 1, p). (13) Suppose p f 1 and ( f 1 ) 1 = p 1 and f is a special sequence and p L( f,1) and h = f 1, [( f 1 ) 1, ( f 1) +p ], p. Then (vi) L(h) = L( f 1) L( f 1, p). (14) Suppose p f 1 and f is a special sequence and i dom f and i+1 dom f and i > 1 and p L( f,i) and p f i and p f i+1 and h = ( f i) p. Then (vi) L(h) = L( f i) L( f i, p). (15) Suppose f f 1 and f is a special sequence and ( f ) = ( f 1 ) and h = f 1,[ ( f 1) 1 +( f ) 1, ( f 1 ) ], f. Then h is a special sequence and h 1 = f 1 and h lenh = f and L(h) is a special polygonal arc joining f 1 and f and L(h) L( f) and L(h) = L( f 1) L( f 1, f ) and L(h) = L( f ) L( f, f ).

3 CONNECTEDNESS CONDITIONS USING POLYGONAL ARCS 3 (16) Suppose f f 1 and f is a special sequence and ( f ) 1 = ( f 1 ) 1 and h = f 1,[( f 1 ) 1, ( f 1 ) +( f ) ], f. Then h is a special sequence and h 1 = f 1 and h lenh = f and L(h) is a special polygonal arc joining f 1 and f and L(h) L( f) and L(h) = L( f 1) L( f 1, f ) and L(h) = L( f ) L( f, f ). (17) Suppose f i f 1 and f is a special sequence and i > and i dom f and h = f i. Then h lenh = f i, (iv) L(h) is a special polygonal arc joining f 1 and f i, (v) L(h) L( f), and (vi) L(h) = L( f i) L( f i, f i ). (18) Suppose p f 1 and f is a special sequence and p L( f,n). Then there exists h such that (v) (vi) L(h) L( f), and L(h) = L( f n) L( f n, p). (19) Suppose p f 1 and f is a special sequence and p L( f). Then there exists h such that h is a special sequence and h 1 = f 1 and h lenh = p and L(h) is a special polygonal arc joining f 1 and p and L(h) L( f). (0) Suppose that p 1 = ( f len f ) 1 and p ( f len f ) or p 1 ( f len f ) 1 and p = ( f len f ) and f 1 / Ball(u,r) and f len f Ball(u,r) and p Ball(u,r) and f is a special sequence andl( f len f, p) L( f) = { f len f } and h = f p. Then h is a special sequence and L(h) is a special polygonal arc joining f 1 and p and L(h) L( f) Ball(u,r). (1) Suppose that f 1 / Ball(u,r) and f len f Ball(u,r) and p Ball(u,r) and [p 1,( f len f ) ] Ball(u,r) and f is a special sequence and p 1 ( f len f ) 1 and p ( f len f ) and h = f [p 1, ( f len f ) ], p and (L( f len f,[p 1,( f len f ) ]) L([p 1,( f len f ) ], p)) L( f) = { f len f }. Then L(h) is a special polygonal arc joining f 1 and p and L(h) L( f) Ball(u,r). () Suppose that f 1 / Ball(u,r) and f len f Ball(u,r) and p Ball(u,r) and [( f len f ) 1, p ] Ball(u,r) and f is a special sequence and p 1 ( f len f ) 1 and p ( f len f ) and h = f [( f len f ) 1, p ], p and (L( f len f,[( f len f ) 1, p ]) L([( f len f ) 1, p ], p)) L( f) = { f len f }. Then L(h) is a special polygonal arc joining f 1 and p and L(h) L( f) Ball(u,r). (3) Suppose f 1 / Ball(u,r) and f len f Ball(u,r) and p Ball(u,r) and f is a special sequence and p / L( f). Then there exists h such that L(h) is a special polygonal arc joining f 1 and p and L(h) L( f) Ball(u,r). (4) Let given R, p, p 1, p, P, r, u. Suppose p p 1 and P is a special polygonal arc joining p 1 and p and P R and p Ball(u,r) and p Ball(u,r) and Ball(u,r) R. Then there exists a subset P 1 ofe T such that P 1 is a special polygonal arc joining p 1 and p and P 1 R. In the sequel P, R are subsets ofe T. We now state several propositions:

4 CONNECTEDNESS CONDITIONS USING POLYGONAL ARCS 4 (5) Let given p. Suppose that and P = {q : q p q R P 1 :subset ofet (P 1 is a special polygonal arc joining p and q P 1 R)}. Then P is open. (6) Suppose that P = {q : q = p P 1 :subset ofe T (P 1 is a special polygonal arc joining p and q P 1 R)}. Then P is open. (7) Suppose p R and P = {q : q = p P 1 :subset ofe T (P 1 is a special polygonal arc joining p and q P 1 R)}. Then P R. (8) Suppose that P = {q : q = p P 1 :subset ofe T (P 1 is a special polygonal arc joining p and q P 1 R)}. Then R P. (9) Suppose that P = {q : q = p P 1 :subset ofe T (P 1 is a special polygonal arc joining p and q P 1 R)}. Then R = P. (30) If R is a region and p R and q R and p q, then there exists P such that P is a special polygonal arc joining p and q and P R. REFERENCES [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, org/jfm/vol1/nat_1.html. [] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, html. [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, [4] Czesław Byliński. Functions and their basic properties. Journal of Formalized Mathematics, 1, funct_1.html. [5] Agata Darmochwał. The Euclidean space. Journal of Formalized Mathematics, 3, [6] Agata Darmochwał and Yatsuka Nakamura. The topological spaceet. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, [7] Agata Darmochwał and Yatsuka Nakamura. The topological spaceet. Simple closed curves. Journal of Formalized Mathematics, 3, [8] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, Vol1/real_1.html. [9] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics,, org/jfm/vol/metric_1.html. [10] Beata Padlewska. Connected spaces. Journal of Formalized Mathematics, 1, [11] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1,

5 CONNECTEDNESS CONDITIONS USING POLYGONAL ARCS 5 [1] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, Axiomatics/tarski.html. [13] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics,, html. [14] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, [15] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, Vol1/relat_1.html. Received August 4, 199 Published January, 004