Article FINTOPO4, MML version 4.99.1005
:: FINTOPO4:prednot 1 => FINTOPO4:pred 1
definition
let a1 be non empty RelStr;
let a2, a3 be Element of bool the carrier of a1;
pred A2,A3 are_separated means
a2 ^b misses a3 & a2 misses a3 ^b;
symmetry;
:: for a1 being non empty RelStr
:: for a2, a3 being Element of bool the carrier of a1
:: st a2,a3 are_separated
:: holds a3,a2 are_separated;
end;
:: FINTOPO4:dfs 1
definiens
let a1 be non empty RelStr;
let a2, a3 be Element of bool the carrier of a1;
To prove
a2,a3 are_separated
it is sufficient to prove
thus a2 ^b misses a3 & a2 misses a3 ^b;
:: FINTOPO4:def 1
theorem
for b1 being non empty RelStr
for b2, b3 being Element of bool the carrier of b1 holds
b2,b3 are_separated
iff
b2 ^b misses b3 & b2 misses b3 ^b;
:: FINTOPO4:th 1
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
st b3 <= b4
holds Finf(b2,b3) c= Finf(b2,b4);
:: FINTOPO4:th 2
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
st b3 <= b4
holds Fcl(b2,b3) c= Fcl(b2,b4);
:: FINTOPO4:th 3
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
st b3 <= b4
holds Fdfl(b2,b4) c= Fdfl(b2,b3);
:: FINTOPO4:th 4
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
st b3 <= b4
holds Fint(b2,b4) c= Fint(b2,b3);
:: FINTOPO4:th 5
theorem
for b1 being non empty RelStr
for b2, b3 being Element of bool the carrier of b1
st b2,b3 are_separated
holds b3,b2 are_separated;
:: FINTOPO4:th 6
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
st b2,b3 are_separated
holds b2 misses b3;
:: FINTOPO4:th 7
theorem
for b1 being non empty RelStr
for b2, b3 being Element of bool the carrier of b1
st b1 is symmetric
holds b2,b3 are_separated
iff
b2 ^f misses b3 & b2 misses b3 ^f;
:: FINTOPO4:th 8
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
st b1 is symmetric & b2 ^b misses b3
holds b2 misses b3 ^b;
:: FINTOPO4:th 9
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
st b1 is symmetric & b2 misses b3 ^b
holds b2 ^b misses b3;
:: FINTOPO4:th 10
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
st b1 is symmetric
holds b2,b3 are_separated
iff
b2 ^b misses b3;
:: FINTOPO4:th 11
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
st b1 is symmetric
holds b2,b3 are_separated
iff
b2 misses b3 ^b;
:: FINTOPO4:th 12
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
st b1 is symmetric
holds b2 is connected(b1)
iff
for b3, b4 being Element of bool the carrier of b1
st b2 = b3 \/ b4 & b3,b4 are_separated & b3 <> b2
holds b4 = b2;
:: FINTOPO4:th 13
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
st b1 is symmetric
holds b2 is connected(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 <> {} & b2 \ b3 <> {} & b3 c= b2
holds b3 ^b meets b2 \ b3;
:: FINTOPO4:prednot 2 => FINTOPO4:pred 2
definition
let a1, a2 be non empty RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be Element of NAT;
pred A3 is_continuous A4 means
for b1 being Element of the carrier of a1
for b2 being Element of the carrier of a2
st b1 in the carrier of a1 & b2 = a3 . b1
holds a3 .: U_FT(b1,0) c= U_FT(b2,a4);
end;
:: FINTOPO4:dfs 2
definiens
let a1, a2 be non empty RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be Element of NAT;
To prove
a3 is_continuous a4
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of the carrier of a2
st b1 in the carrier of a1 & b2 = a3 . b1
holds a3 .: U_FT(b1,0) c= U_FT(b2,a4);
:: FINTOPO4:def 2
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 holds
b3 is_continuous b4
iff
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
st b5 in the carrier of b1 & b6 = b3 . b5
holds b3 .: U_FT(b5,0) c= U_FT(b6,b4);
:: FINTOPO4:th 14
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive RelStr
for b3 being Element of NAT
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b4 is_continuous 0
holds b4 is_continuous b3;
:: FINTOPO4:th 15
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive RelStr
for b3, b4 being Element of NAT
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b5 is_continuous b3 & b3 <= b4
holds b5 is_continuous b4;
:: FINTOPO4:th 16
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b5 is_continuous 0 & b4 = b5 .: b3
holds b5 .: (b3 ^b) c= b4 ^b;
:: FINTOPO4:th 17
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is connected(b1) & b5 is_continuous 0 & b4 = b5 .: b3
holds b4 is connected(b2);
:: FINTOPO4:funcnot 1 => FINTOPO4:func 1
definition
let a1 be natural set;
func Nbdl1 A1 -> Relation of Seg a1,Seg a1 means
for b1 being Element of NAT
st b1 in Seg a1
holds Im(it,b1) = {b1,max(b1 -' 1,1),min(b1 + 1,a1)};
end;
:: FINTOPO4:def 3
theorem
for b1 being natural set
for b2 being Relation of Seg b1,Seg b1 holds
b2 = Nbdl1 b1
iff
for b3 being Element of NAT
st b3 in Seg b1
holds Im(b2,b3) = {b3,max(b3 -' 1,1),min(b3 + 1,b1)};
:: FINTOPO4:funcnot 2 => FINTOPO4:func 2
definition
let a1 be natural set;
assume 0 < a1;
func FTSL1 A1 -> non empty RelStr equals
RelStr(#Seg a1,Nbdl1 a1#);
end;
:: FINTOPO4:def 4
theorem
for b1 being natural set
st 0 < b1
holds FTSL1 b1 = RelStr(#Seg b1,Nbdl1 b1#);
:: FINTOPO4:th 18
theorem
for b1 being natural set
st 0 < b1
holds FTSL1 b1 is reflexive;
:: FINTOPO4:th 19
theorem
for b1 being natural set
st 0 < b1
holds FTSL1 b1 is symmetric;
:: FINTOPO4:funcnot 3 => FINTOPO4:func 3
definition
let a1 be natural set;
func Nbdc1 A1 -> Relation of Seg a1,Seg a1 means
for b1 being Element of NAT
st b1 in Seg a1
holds (1 < b1 & b1 < a1 implies Im(it,b1) = {b1,b1 -' 1,b1 + 1}) &
(b1 = 1 & b1 < a1 implies Im(it,b1) = {b1,a1,b1 + 1}) &
(1 < b1 & b1 = a1 implies Im(it,b1) = {b1,b1 -' 1,1}) &
(b1 = 1 & b1 = a1 implies Im(it,b1) = {b1});
end;
:: FINTOPO4:def 5
theorem
for b1 being natural set
for b2 being Relation of Seg b1,Seg b1 holds
b2 = Nbdc1 b1
iff
for b3 being Element of NAT
st b3 in Seg b1
holds (1 < b3 & b3 < b1 implies Im(b2,b3) = {b3,b3 -' 1,b3 + 1}) &
(b3 = 1 & b3 < b1 implies Im(b2,b3) = {b3,b1,b3 + 1}) &
(1 < b3 & b3 = b1 implies Im(b2,b3) = {b3,b3 -' 1,1}) &
(b3 = 1 & b3 = b1 implies Im(b2,b3) = {b3});
:: FINTOPO4:funcnot 4 => FINTOPO4:func 4
definition
let a1 be natural set;
assume 0 < a1;
func FTSC1 A1 -> non empty RelStr equals
RelStr(#Seg a1,Nbdc1 a1#);
end;
:: FINTOPO4:def 6
theorem
for b1 being natural set
st 0 < b1
holds FTSC1 b1 = RelStr(#Seg b1,Nbdc1 b1#);
:: FINTOPO4:th 20
theorem
for b1 being Element of NAT
st 0 < b1
holds FTSC1 b1 is reflexive;
:: FINTOPO4:th 21
theorem
for b1 being Element of NAT
st 0 < b1
holds FTSC1 b1 is symmetric;