Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### The Jordan's Property for Certain Subsets of the Plane

**by****Yatsuka Nakamura, and****Jaroslaw Kotowicz**- Received August 24, 1992
- MML identifier: JORDAN1

- [ Mizar article, MML identifier index ]

environ vocabulary PRE_TOPC, FUNCT_1, RELAT_1, SUBSET_1, RELAT_2, ORDINAL2, BOOLE, RCOMP_1, BORSUK_1, FUNCOP_1, CONNSP_1, EUCLID, TOPREAL1, TOPS_2, ARYTM_1, MCART_1, ARYTM_3, SQUARE_1, PCOMPS_1, METRIC_1, JORDAN1, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, XREAL_0, REAL_1, RELAT_1, NAT_1, FUNCT_1, FUNCT_2, BINOP_1, PRE_TOPC, TOPS_2, CONNSP_1, SQUARE_1, RCOMP_1, STRUCT_0, METRIC_1, PCOMPS_1, BORSUK_1, TOPREAL1, EUCLID; constructors REAL_1, TOPS_2, CONNSP_1, SQUARE_1, RCOMP_1, TOPMETR, TOPREAL1, MEMBERED, NAT_1; clusters PRE_TOPC, BORSUK_1, STRUCT_0, EUCLID, TOPREAL1, XREAL_0, RELSET_1, MEMBERED, ZFMISC_1; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin :: :: Selected theorems on connected space :: reserve GX,GY for non empty TopSpace, x,y for Point of GX, r,s for Real; theorem :: JORDAN1:1 for GX being TopStruct, A being Subset of GX holds the carrier of (GX|A) = A; theorem :: JORDAN1:2 for GX being non empty TopSpace st (for x,y being Point of GX ex GY st (GY is connected & ex f being map of GY,GX st f is continuous & x in rng(f)& y in rng(f))) holds GX is connected; canceled; theorem :: JORDAN1:4 ::Arcwise connectedness derives connectedness for GX being non empty TopSpace st (for x,y being Point of GX ex h being map of I[01],GX st h is continuous & x=h.0 & y=h.1) holds GX is connected; theorem :: JORDAN1:5 ::Arcwise connectedness derives connectedness for subset for A being Subset of GX st (for xa,ya being Point of GX st xa in A & ya in A & xa<>ya ex h being map of I[01],GX|A st h is continuous & xa=h.0 & ya=h.1) holds A is connected; theorem :: JORDAN1:6 for A0 being Subset of GX, A1 being Subset of GX st A0 is connected & A1 is connected & A0 meets A1 holds A0 \/ A1 is connected; theorem :: JORDAN1:7 for A0,A1,A2 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A0 meets A1 & A1 meets A2 holds A0 \/ A1 \/ A2 is connected; theorem :: JORDAN1:8 for A0,A1,A2,A3 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A3 is connected & A0 meets A1 & A1 meets A2 & A2 meets A3 holds A0 \/ A1 \/ A2 \/ A3 is connected; begin :: :: Certain connected and open subsets in the Euclidean plane :: reserve Q,P1,P2 for Subset of TOP-REAL 2; reserve P for Subset of TOP-REAL 2; reserve w1,w2 for Point of TOP-REAL 2; definition let n be Nat, P be Subset of TOP-REAL n; attr P is convex means :: JORDAN1:def 1 for w1,w2 being Point of TOP-REAL n st w1 in P & w2 in P holds LSeg(w1,w2) c= P; end; theorem :: JORDAN1:9 for n being Nat, P being Subset of TOP-REAL n st P is convex holds P is connected; reserve pa,pb for Point of TOP-REAL 2, s1,t1,s2,t2,s,t,s3,t3,s4,t4,s5,t5,s6,t6, l,lambda,sa,sd,ta,td for Real; theorem :: JORDAN1:10 s1<s3 & s1<s4 & 0 <= l & l <= 1 implies s1<(1-l)*s3+l*s4; theorem :: JORDAN1:11 s3<s1 & s4<s1 & 0 <= l & l <= 1 implies (1-l)*s3+l*s4<s1; reserve s1a,t1a,s2a,t2a,s3a,t3a,sb,tb,sc,tc for Real; theorem :: JORDAN1:12 {|[ s,t ]|: s1<s & s<s2 & t1<t & t<t2} = {|[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2} /\ {|[s5,t5]|:t1<t5} /\ {|[s6,t6]|:t6<t2}; theorem :: JORDAN1:13 {|[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} = {|[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1} \/ {|[s5,t5]|:s2<s5} \/ {|[s6,t6]|:t2<t6}; theorem :: JORDAN1:14 for s1,t1,s2,t2,P st P = {|[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} holds P is convex; theorem :: JORDAN1:15 for s1,t1,s2,t2,P st P = {|[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} holds P is connected; theorem :: JORDAN1:16 for s1,P st P = { |[ s,t ]|:s1<s } holds P is convex; theorem :: JORDAN1:17 for s1,P st P = { |[ s,t ]|:s1<s } holds P is connected; theorem :: JORDAN1:18 for s2,P st P = {|[ s,t ]|:s<s2 } holds P is convex; theorem :: JORDAN1:19 for s2,P st P = {|[ s,t ]|:s<s2 } holds P is connected; theorem :: JORDAN1:20 for t1,P st P = {|[ s,t ]|:t1<t } holds P is convex; theorem :: JORDAN1:21 for t1,P st P = {|[ s,t ]|:t1<t } holds P is connected; theorem :: JORDAN1:22 for t2,P st P = { |[ s,t ]|: t<t2 } holds P is convex; theorem :: JORDAN1:23 for t2,P st P = { |[ s,t ]|: t<t2 } holds P is connected; theorem :: JORDAN1:24 for s1,t1,s2,t2,P st P = { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} holds P is connected; theorem :: JORDAN1:25 for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1<s } holds P is open; theorem :: JORDAN1:26 for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1>s } holds P is open; theorem :: JORDAN1:27 for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1<t } holds P is open; theorem :: JORDAN1:28 for s1 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1>t } holds P is open; theorem :: JORDAN1:29 for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} holds P is open; theorem :: JORDAN1:30 for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} holds P is open; theorem :: JORDAN1:31 for s1,t1,s2,t2,P,Q st P = { |[ sa,ta ]|:s1<sa & sa<s2 & t1<ta & ta<t2} & Q = { |[ sb,tb ]|:not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} holds P misses Q; theorem :: JORDAN1:32 for s1,s2,t1,t2 be Real holds {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2} = {|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2}; theorem :: JORDAN1:33 for s1,s2,t1,t2 holds {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} = {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)}; theorem :: JORDAN1:34 for s1,s2,t1,t2 holds { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2} is Subset of TOP-REAL 2; theorem :: JORDAN1:35 for s1,s2,t1,t2 holds { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)} is Subset of TOP-REAL 2; theorem :: JORDAN1:36 for s1,t1,s2,t2,P st P = { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2} holds P is connected; theorem :: JORDAN1:37 for s1,t1,s2,t2,P st P = { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)} holds P is connected; theorem :: JORDAN1:38 for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { p0 where p0 is Point of TOP-REAL 2:s1<p0`1 & p0`1<s2 & t1<p0`2 & p0`2<t2} holds P is open; theorem :: JORDAN1:39 for s1,t1,s2,t2 for P being Subset of TOP-REAL 2 st P = { pq where pq is Point of TOP-REAL 2: not (s1<=pq`1 & pq`1<=s2 & t1<=pq`2 & pq`2<=t2)} holds P is open; theorem :: JORDAN1:40 for s1,t1,s2,t2,P,Q st P = {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2} & Q = {qc where qc is Point of TOP-REAL 2: not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} holds P misses Q; theorem :: JORDAN1:41 for s1,t1,s2,t2 for P,P1,P2 being Subset of TOP-REAL 2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P`= P1 \/ P2 & P`<> {} & P1 misses P2 & for P1A,P2B being Subset of (TOP-REAL 2)|P` st P1A = P1 & P2B = P2 holds P1A is_a_component_of (TOP-REAL 2)|P` & P2B is_a_component_of (TOP-REAL 2)|P` ; theorem :: JORDAN1:42 for s1,t1,s2,t2 for P,P1,P2 being Subset of TOP-REAL 2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P = (Cl P1) \ P1 & P = (Cl P2) \P2; theorem :: JORDAN1:43 for s1,s2,t1,t2,P,P1 st P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} holds P1 c= [#] ((TOP-REAL 2)|P`); theorem :: JORDAN1:44 for s1,s2,t1,t2,P,P1 st P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2} holds P1 is Subset of (TOP-REAL 2)|P`; theorem :: JORDAN1:45 for s1,s2,t1,t2,P,P2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P2 c= [#]((TOP-REAL 2)|P`); theorem :: JORDAN1:46 for s1,s2,t1,t2,P,P2 st s1<s2 & t1< t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} & P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds P2 is Subset of (TOP-REAL 2)|P`; begin :: :: The Jordan's property :: definition let S be Subset of TOP-REAL 2; attr S is Jordan means :: JORDAN1:def 2 S`<> {} & ex A1,A2 being Subset of TOP-REAL 2 st S` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & for C1,C2 being Subset of (TOP-REAL 2)|S` st C1 = A1 & C2 = A2 holds C1 is_a_component_of (TOP-REAL 2)|S` & C2 is_a_component_of (TOP-REAL 2)|S`; synonym S has_property_J; end; theorem :: JORDAN1:47 for S being Subset of TOP-REAL 2 st S has_property_J holds S`<> {} & ex A1,A2 being Subset of TOP-REAL 2 st ex C1,C2 being Subset of (TOP-REAL 2)|S` st S` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & C1 = A1 & C2 = A2 & C1 is_a_component_of (TOP-REAL 2)|S` & C2 is_a_component_of (TOP-REAL 2)|S` & for C3 being Subset of (TOP-REAL 2)|S` st C3 is_a_component_of (TOP-REAL 2)|S` holds C3 = C1 or C3 = C2; theorem :: JORDAN1:48 for s1,s2,t1,t2 for P being Subset of TOP-REAL 2 st s1<s2 & t1<t2 & P = { p where p is Point of TOP-REAL 2: p`1 = s1 & p`2 <= t2 & p`2 >= t1 or p`1 <= s2 & p`1 >= s1 & p`2 = t2 or p`1 <= s2 & p`1 >= s1 & p`2 = t1 or p`1 = s2 & p`2 <= t2 & p`2 >= t1} holds P has_property_J;

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