:: Homeomorphism between Finite Topological Spaces, Two-Dimensional LatticeSpaces and a Fixed Point Theorem
:: by Masami Tanaka , Hiroshi Imura and Yatsuka Nakamura
::
:: Received July 6, 2005
:: Copyright (c) 2005 Association of Mizar Users


theorem Th1: :: FINTOPO5:1
for X being set
for Y being non empty set
for f being Function of X,Y
for A being Subset of X st f is one-to-one holds
(f " ) .: (f .: A) = A
proof end;

theorem :: FINTOPO5:2
for n being Element of NAT holds
( n > 0 iff Seg n <> {} ) by FINSEQ_1:4;

definition
let FT1, FT2 be RelStr ;
let h be Function of FT1,FT2;
attr h is being_homeomorphism means :Def1: :: FINTOPO5:def 1
( h is one-to-one & h is onto & ( for x being Element of FT1 holds h .: (U_FT x) = Im the InternalRel of FT2,(h . x) ) );
end;

:: deftheorem Def1 defines being_homeomorphism FINTOPO5:def 1 :
for FT1, FT2 being RelStr
for h being Function of FT1,FT2 holds
( h is being_homeomorphism iff ( h is one-to-one & h is onto & ( for x being Element of FT1 holds h .: (U_FT x) = Im the InternalRel of FT2,(h . x) ) ) );

theorem Th3: :: FINTOPO5:3
for FT1, FT2 being non empty RelStr
for h being Function of FT1,FT2 st h is being_homeomorphism holds
ex g being Function of FT2,FT1 st
( g = h " & g is being_homeomorphism )
proof end;

theorem Th4: :: FINTOPO5:4
for FT1, FT2 being non empty RelStr
for h being Function of FT1,FT2
for n being Element of NAT
for x being Element of FT1
for y being Element of FT2 st h is being_homeomorphism & y = h . x holds
for z being Element of FT1 holds
( z in U_FT x,n iff h . z in U_FT y,n )
proof end;

theorem :: FINTOPO5:5
for FT1, FT2 being non empty RelStr
for h being Function of FT1,FT2
for n being Element of NAT
for x being Element of FT1
for y being Element of FT2 st h is being_homeomorphism & y = h . x holds
for v being Element of FT2 holds
( (h " ) . v in U_FT x,n iff v in U_FT y,n )
proof end;

theorem :: FINTOPO5:6
for n being non zero Element of NAT
for f being Function of (FTSL1 n),(FTSL1 n) st f is_continuous 0 holds
ex p being Element of (FTSL1 n) st f . p in U_FT p,0
proof end;

theorem Th7: :: FINTOPO5:7
for T being non empty RelStr
for p being Element of T
for k being Element of NAT st T is filled holds
U_FT p,k c= U_FT p,(k + 1)
proof end;

theorem Th8: :: FINTOPO5:8
for T being non empty RelStr
for p being Element of T
for k being Element of NAT st T is filled holds
U_FT p,0 c= U_FT p,k
proof end;

theorem Th9: :: FINTOPO5:9
for n being non zero Nat
for jn, j, k being Nat
for p being Element of (FTSL1 n) st p = jn holds
( j in U_FT p,k iff ( j in Seg n & abs (jn - j) <= k + 1 ) )
proof end;

theorem :: FINTOPO5:10
for kc, km being Element of NAT
for n being non zero Element of NAT
for f being Function of (FTSL1 n),(FTSL1 n) st f is_continuous kc & km = [/(kc / 2)\] holds
ex p being Element of (FTSL1 n) st f . p in U_FT p,km
proof end;

definition
let A, B be set ;
let R be Relation of A,B;
let x be set ;
:: original: Im
redefine func Im R,x -> Subset of B;
coherence
Im R,x is Subset of B
proof end;
end;

definition
let n, m be Element of NAT ;
func Nbdl2 n,m -> Relation of [:(Seg n),(Seg m):] means :Def2: :: FINTOPO5:def 2
for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im it,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):];
existence
ex b1 being Relation of [:(Seg n),(Seg m):] st
for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b1,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):]
proof end;
uniqueness
for b1, b2 being Relation of [:(Seg n),(Seg m):] st ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b1,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] ) & ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b2,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines Nbdl2 FINTOPO5:def 2 :
for n, m being Element of NAT
for b3 being Relation of [:(Seg n),(Seg m):] holds
( b3 = Nbdl2 n,m iff for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b3,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] );

definition
let n, m be Element of NAT ;
func FTSL2 n,m -> strict RelStr equals :: FINTOPO5:def 3
 RelStr(# [:(Seg n),(Seg m):],(Nbdl2 n,m) #);
coherence
RelStr(# [:(Seg n),(Seg m):],(Nbdl2 n,m) #) is strict RelStr ;
end;

:: deftheorem defines FTSL2 FINTOPO5:def 3 :
for n, m being Element of NAT holds FTSL2 n,m = RelStr(# [:(Seg n),(Seg m):],(Nbdl2 n,m) #);

registration
let n, m be non zero Element of NAT ;
cluster FTSL2 n,m -> non empty strict ;
coherence
not FTSL2 n,m is empty ;
end;

theorem :: FINTOPO5:11
for n, m being non zero Element of NAT holds FTSL2 n,m is filled
proof end;

theorem :: FINTOPO5:12
for n, m being non zero Element of NAT holds FTSL2 n,m is symmetric
proof end;

theorem :: FINTOPO5:13
for n being non zero Element of NAT ex h being Function of (FTSL2 n,1),(FTSL1 n) st h is being_homeomorphism
proof end;

definition
let n, m be Element of NAT ;
func Nbds2 n,m -> Relation of [:(Seg n),(Seg m):] means :Def4: :: FINTOPO5:def 4
for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im it,x = [:{i},(Im (Nbdl1 m),j):] \/ [:(Im (Nbdl1 n),i),{j}:];
existence
ex b1 being Relation of [:(Seg n),(Seg m):] st
for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b1,x = [:{i},(Im (Nbdl1 m),j):] \/ [:(Im (Nbdl1 n),i),{j}:]
proof end;
uniqueness
for b1, b2 being Relation of [:(Seg n),(Seg m):] st ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b1,x = [:{i},(Im (Nbdl1 m),j):] \/ [:(Im (Nbdl1 n),i),{j}:] ) & ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b2,x = [:{i},(Im (Nbdl1 m),j):] \/ [:(Im (Nbdl1 n),i),{j}:] ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines Nbds2 FINTOPO5:def 4 :
for n, m being Element of NAT
for b3 being Relation of [:(Seg n),(Seg m):] holds
( b3 = Nbds2 n,m iff for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im b3,x = [:{i},(Im (Nbdl1 m),j):] \/ [:(Im (Nbdl1 n),i),{j}:] );

definition
let n, m be Element of NAT ;
func FTSS2 n,m -> strict RelStr equals :: FINTOPO5:def 5
 RelStr(# [:(Seg n),(Seg m):],(Nbds2 n,m) #);
coherence
RelStr(# [:(Seg n),(Seg m):],(Nbds2 n,m) #) is strict RelStr ;
end;

:: deftheorem defines FTSS2 FINTOPO5:def 5 :
for n, m being Element of NAT holds FTSS2 n,m = RelStr(# [:(Seg n),(Seg m):],(Nbds2 n,m) #);

registration
let n, m be non zero Element of NAT ;
cluster FTSS2 n,m -> non empty strict ;
coherence
not FTSS2 n,m is empty ;
end;

theorem :: FINTOPO5:14
for n, m being non zero Element of NAT holds FTSS2 n,m is filled
proof end;

theorem :: FINTOPO5:15
for n, m being non zero Element of NAT holds FTSS2 n,m is symmetric
proof end;

theorem :: FINTOPO5:16
for n being non zero Element of NAT ex h being Function of (FTSS2 n,1),(FTSL1 n) st h is being_homeomorphism
proof end;