Article FINTOPO5, MML version 4.99.1005
:: FINTOPO5:th 1
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b1
st b3 is one-to-one
holds b3 " .: (b3 .: b4) = b4;
:: FINTOPO5:th 2
theorem
for b1 being Element of NAT holds
0 < b1
iff
Seg b1 <> {};
:: FINTOPO5:attrnot 1 => FINTOPO5:attr 1
definition
let a1, a2 be RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is being_homeomorphism means
a3 is one-to-one &
a3 is onto(the carrier of a1, the carrier of a2) &
(for b1 being Element of the carrier of a1 holds
a3 .: U_FT b1 = Im(the InternalRel of a2,a3 . b1));
end;
:: FINTOPO5:dfs 1
definiens
let a1, a2 be RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is being_homeomorphism
it is sufficient to prove
thus a3 is one-to-one &
a3 is onto(the carrier of a1, the carrier of a2) &
(for b1 being Element of the carrier of a1 holds
a3 .: U_FT b1 = Im(the InternalRel of a2,a3 . b1));
:: FINTOPO5:def 1
theorem
for b1, b2 being RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is being_homeomorphism(b1, b2)
iff
b3 is one-to-one &
b3 is onto(the carrier of b1, the carrier of b2) &
(for b4 being Element of the carrier of b1 holds
b3 .: U_FT b4 = Im(the InternalRel of b2,b3 . b4));
:: FINTOPO5:th 3
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is being_homeomorphism(b1, b2)
holds ex b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
b4 = b3 " & b4 is being_homeomorphism(b2, b1);
:: FINTOPO5:th 4
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of NAT
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
st b3 is being_homeomorphism(b1, b2) & b6 = b3 . b5
for b7 being Element of the carrier of b1 holds
b7 in U_FT(b5,b4)
iff
b3 . b7 in U_FT(b6,b4);
:: FINTOPO5:th 5
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of NAT
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
st b3 is being_homeomorphism(b1, b2) & b6 = b3 . b5
for b7 being Element of the carrier of b2 holds
b3 " . b7 in U_FT(b5,b4)
iff
b7 in U_FT(b6,b4);
:: FINTOPO5:th 6
theorem
for b1 being non empty Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of FTSL1 b1,the carrier of FTSL1 b1
st b2 is_continuous 0
holds ex b3 being Element of the carrier of FTSL1 b1 st
b2 . b3 in U_FT(b3,0);
:: FINTOPO5:th 7
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT
st b1 is reflexive
holds U_FT(b2,b3) c= U_FT(b2,b3 + 1);
:: FINTOPO5:th 8
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT
st b1 is reflexive
holds U_FT(b2,0) c= U_FT(b2,b3);
:: FINTOPO5:th 9
theorem
for b1 being non empty natural set
for b2, b3, b4 being natural set
for b5 being Element of the carrier of FTSL1 b1
st b5 = b2
holds b3 in U_FT(b5,b4)
iff
b3 in Seg b1 & abs (b2 - b3) <= b4 + 1;
:: FINTOPO5:th 10
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of NAT
for b4 being Function-like quasi_total Relation of the carrier of FTSL1 b3,the carrier of FTSL1 b3
st b4 is_continuous b1 & b2 = [/b1 / 2\]
holds ex b5 being Element of the carrier of FTSL1 b3 st
b4 . b5 in U_FT(b5,b2);
:: FINTOPO5:funcnot 1 => FINTOPO5:func 1
definition
let a1, a2 be set;
let a3 be Relation of a1,a2;
let a4 be set;
redefine func Im(a3,a4) -> Element of bool a2;
end;
:: FINTOPO5:funcnot 2 => FINTOPO5:func 2
definition
let a1, a2 be Element of NAT;
func Nbdl2(A1,A2) -> Relation of [:Seg a1,Seg a2:],[:Seg a1,Seg a2:] means
for b1 being set
st b1 in [:Seg a1,Seg a2:]
for b2, b3 being Element of NAT
st b1 = [b2,b3]
holds Im(it,b1) = [:Im(Nbdl1 a1,b2),Im(Nbdl1 a2,b3):];
end;
:: FINTOPO5:def 2
theorem
for b1, b2 being Element of NAT
for b3 being Relation of [:Seg b1,Seg b2:],[:Seg b1,Seg b2:] holds
b3 = Nbdl2(b1,b2)
iff
for b4 being set
st b4 in [:Seg b1,Seg b2:]
for b5, b6 being Element of NAT
st b4 = [b5,b6]
holds Im(b3,b4) = [:Im(Nbdl1 b1,b5),Im(Nbdl1 b2,b6):];
:: FINTOPO5:funcnot 3 => FINTOPO5:func 3
definition
let a1, a2 be Element of NAT;
func FTSL2(A1,A2) -> strict RelStr equals
RelStr(#[:Seg a1,Seg a2:],Nbdl2(a1,a2)#);
end;
:: FINTOPO5:def 3
theorem
for b1, b2 being Element of NAT holds
FTSL2(b1,b2) = RelStr(#[:Seg b1,Seg b2:],Nbdl2(b1,b2)#);
:: FINTOPO5:funcreg 1
registration
let a1, a2 be non empty Element of NAT;
cluster FTSL2(a1,a2) -> non empty strict;
end;
:: FINTOPO5:th 11
theorem
for b1, b2 being non empty Element of NAT holds
FTSL2(b1,b2) is reflexive;
:: FINTOPO5:th 12
theorem
for b1, b2 being non empty Element of NAT holds
FTSL2(b1,b2) is symmetric;
:: FINTOPO5:th 13
theorem
for b1 being non empty Element of NAT holds
ex b2 being Function-like quasi_total Relation of the carrier of FTSL2(b1,1),the carrier of FTSL1 b1 st
b2 is being_homeomorphism(FTSL2(b1,1), FTSL1 b1);
:: FINTOPO5:funcnot 4 => FINTOPO5:func 4
definition
let a1, a2 be Element of NAT;
func Nbds2(A1,A2) -> Relation of [:Seg a1,Seg a2:],[:Seg a1,Seg a2:] means
for b1 being set
st b1 in [:Seg a1,Seg a2:]
for b2, b3 being Element of NAT
st b1 = [b2,b3]
holds Im(it,b1) = [:{b2},Im(Nbdl1 a2,b3):] \/ [:Im(Nbdl1 a1,b2),{b3}:];
end;
:: FINTOPO5:def 4
theorem
for b1, b2 being Element of NAT
for b3 being Relation of [:Seg b1,Seg b2:],[:Seg b1,Seg b2:] holds
b3 = Nbds2(b1,b2)
iff
for b4 being set
st b4 in [:Seg b1,Seg b2:]
for b5, b6 being Element of NAT
st b4 = [b5,b6]
holds Im(b3,b4) = [:{b5},Im(Nbdl1 b2,b6):] \/ [:Im(Nbdl1 b1,b5),{b6}:];
:: FINTOPO5:funcnot 5 => FINTOPO5:func 5
definition
let a1, a2 be Element of NAT;
func FTSS2(A1,A2) -> strict RelStr equals
RelStr(#[:Seg a1,Seg a2:],Nbds2(a1,a2)#);
end;
:: FINTOPO5:def 5
theorem
for b1, b2 being Element of NAT holds
FTSS2(b1,b2) = RelStr(#[:Seg b1,Seg b2:],Nbds2(b1,b2)#);
:: FINTOPO5:funcreg 2
registration
let a1, a2 be non empty Element of NAT;
cluster FTSS2(a1,a2) -> non empty strict;
end;
:: FINTOPO5:th 14
theorem
for b1, b2 being non empty Element of NAT holds
FTSS2(b1,b2) is reflexive;
:: FINTOPO5:th 15
theorem
for b1, b2 being non empty Element of NAT holds
FTSS2(b1,b2) is symmetric;
:: FINTOPO5:th 16
theorem
for b1 being non empty Element of NAT holds
ex b2 being Function-like quasi_total Relation of the carrier of FTSS2(b1,1),the carrier of FTSL1 b1 st
b2 is being_homeomorphism(FTSS2(b1,1), FTSL1 b1);