:: Homeomorphism between Finite Topological Spaces, Two-Dimensional Lattice
:: Spaces and a Fixed Point Theorem
:: by Masami Tanaka , Hiroshi Imura and Yatsuka Nakamura
::
:: Received July 6, 2005
:: Copyright (c) 2005-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, XBOOLE_0, FUNCT_1, SUBSET_1, RELAT_1, XXREAL_0, CARD_1,
FINSEQ_1, ORDERS_2, TOPS_2, FUNCT_2, FIN_TOPO, STRUCT_0, ARYTM_3,
EQREL_1, XCMPLX_0, FINTOPO4, NAT_1, ARYTM_1, TARSKI, COMPLEX1, INT_1,
ZFMISC_1, RELAT_2, FINTOPO5;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,
XXREAL_0, NAT_1, NAT_D, RELAT_1, FUNCT_1, RELSET_1, EQREL_1, FUNCT_2,
FINSEQ_1, STRUCT_0, FIN_TOPO, FINTOPO3, ENUMSET1, ORDERS_2, FINTOPO4,
COMPLEX1, INT_1;
constructors ENUMSET1, REAL_1, NAT_1, EQREL_1, FIN_TOPO, FINTOPO3, FINTOPO4,
NAT_D, RELSET_1;
registrations XBOOLE_0, SUBSET_1, RELSET_1, XXREAL_0, XREAL_0, NAT_1, INT_1,
STRUCT_0, FIN_TOPO, ORDINAL1, FINSEQ_1, CARD_1;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
begin
theorem :: FINTOPO5:1
for X being set, Y being non empty set, f being Function of X,Y,
A being Subset of X st f is one-to-one holds f".:(f.:A)=A;
theorem :: FINTOPO5:2
for n being Nat holds n>0 iff Seg n<>{};
definition
let FT1,FT2 be RelStr, h be Function of FT1, FT2;
attr h is being_homeomorphism means
:: FINTOPO5:def 1
h is one-to-one onto & for x
being Element of FT1 holds h.:U_FT x = Im(the InternalRel of FT2,h.x);
end;
theorem :: FINTOPO5:3
for FT1,FT2 being non empty RelStr, h being Function of FT1, FT2
st h is being_homeomorphism ex g being Function of FT2, FT1 st g=h" & g is
being_homeomorphism;
theorem :: FINTOPO5:4
for FT1,FT2 being non empty RelStr, h being Function of FT1, FT2,
n being Nat, x being Element of FT1, 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);
theorem :: FINTOPO5:5
for FT1,FT2 being non empty RelStr, h being Function of FT1, FT2, n
being Nat,x being Element of FT1,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);
theorem :: FINTOPO5:6
for n being non zero Nat, 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);
theorem :: FINTOPO5:7
for T being non empty RelStr, p being Element of T, k being
Nat st T is filled holds U_FT(p,k) c= U_FT(p,k+1);
theorem :: FINTOPO5:8
for T being non empty RelStr, p being Element of T, k being
Nat st T is filled holds U_FT(p,0) c= U_FT(p,k);
theorem :: FINTOPO5:9
for n being non zero Nat, jn,j,k being Nat, p being Element of
FTSL1 n st p=jn holds j in U_FT(p,k) iff j in Seg n & |.jn-j.|<= k+1;
:: Fixed Point Theorem
theorem :: FINTOPO5:10
for kc,km being Nat, n being non zero Nat, 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);
definition
let A,B be set;
let R be Relation of A,B, x be set;
redefine func Im(R,x) -> Subset of B;
end;
:: 2-dimensional linear FT_Str
definition
let n,m be Nat;
func Nbdl2(n,m) -> Relation of [:Seg n, Seg m:] means
:: FINTOPO5:def 2
for x being set
st x in [:Seg n, Seg m:] holds for i,j being Nat st x=[i,j] holds Im
(it,x) = [:Im(Nbdl1 n,i), Im(Nbdl1 m,j):];
end;
definition
let n,m be Nat;
func FTSL2(n,m) -> strict RelStr equals
:: FINTOPO5:def 3
RelStr(# [:Seg n, Seg m:], Nbdl2(n,m
) #);
end;
registration
let n,m be non zero Nat;
cluster FTSL2(n,m) -> non empty;
end;
theorem :: FINTOPO5:11
for n,m being non zero Nat holds FTSL2(n,m) is filled;
theorem :: FINTOPO5:12
for n,m being non zero Nat holds FTSL2(n,m) is symmetric;
theorem :: FINTOPO5:13
for n being non zero Nat ex h being Function of FTSL2(n,1),
FTSL1 n st h is being_homeomorphism;
:: 2-dimensional small FT_Str
definition
let n,m be Nat;
func Nbds2(n,m) -> Relation of [:Seg n, Seg m:] means
:: FINTOPO5:def 4
for x being set
st x in [:Seg n, Seg m:] holds for i,j being Nat st x=[i,j] holds Im
(it,x) = [:{i}, Im(Nbdl1 m,j):] \/ [:Im(Nbdl1 n,i),{j}:];
end;
definition
let n,m be Nat;
func FTSS2(n,m) -> strict RelStr equals
:: FINTOPO5:def 5
RelStr(# [:Seg n, Seg m:], Nbds2(n,m
) #);
end;
registration
let n,m be non zero Nat;
cluster FTSS2(n,m) -> non empty;
end;
theorem :: FINTOPO5:14
for n,m being non zero Nat holds FTSS2(n,m) is filled;
theorem :: FINTOPO5:15
for n,m being non zero Nat holds FTSS2(n,m) is symmetric;
theorem :: FINTOPO5:16
for n being non zero Nat ex h being Function of FTSS2(n,1),
FTSL1 n st h is being_homeomorphism;