Article TOPGEN_5, MML version 4.99.1005
:: TOPGEN_5:th 1
theorem
for b1, b2 being Relation-like Function-like set
st b1 tolerates b2
for b3 being set holds
(b1 +* b2) " b3 = (b1 " b3) \/ (b2 " b3);
:: TOPGEN_5:th 2
theorem
for b1, b2 being Relation-like Function-like set
st proj1 b1 misses proj1 b2
for b3 being set holds
(b1 +* b2) " b3 = (b1 " b3) \/ (b2 " b3);
:: TOPGEN_5:th 3
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
st b2 in proj1 b3
holds (commute (b1 .--> b3)) . b2 = b1 .--> (b3 . b2);
:: TOPGEN_5:th 4
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
b1 in proj1 commute b2
iff
ex b3 being set st
ex b4 being Relation-like Function-like set st
b3 in proj1 b2 & b4 = b2 . b3 & b1 in proj1 b4;
:: TOPGEN_5:th 5
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
b1 in proj1 ((commute b3) . b2)
iff
ex b4 being Relation-like Function-like set st
b1 in proj1 b3 & b4 = b3 . b1 & b2 in proj1 b4;
:: TOPGEN_5:th 6
theorem
for b1, b2 being set
for b3, b4 being Relation-like Function-like set
st b1 in proj1 b3 & b4 = b3 . b1 & b2 in proj1 b4
holds ((commute b3) . b2) . b1 = b4 . b2;
:: TOPGEN_5:th 7
theorem
for b1 being set
for b2, b3, b4 being Relation-like Function-like set
st b4 = b2 \/ b3
holds (commute b4) . b1 = ((commute b2) . b1) \/ ((commute b3) . b1);
:: TOPGEN_5:condreg 1
registration
cluster finite -> bounded (Element of bool REAL);
end;
:: TOPGEN_5:th 8
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 <= b4
holds ].b1,b3.[ /\ [.b2,b4.] = [.b2,b3.[;
:: TOPGEN_5:th 9
theorem
for b1, b2, b3, b4 being real set
st b2 <= b1 & b4 < b3
holds ].b1,b3.[ /\ [.b2,b4.] = ].b1,b4.];
:: TOPGEN_5:th 10
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b2 < b3 & b3 <= b4
holds [.b1,b3.[ \/ ].b2,b4.] = [.b1,b4.];
:: TOPGEN_5:th 11
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b2 < b3 & b3 <= b4
holds [.b1,b3.[ /\ ].b2,b4.] = ].b2,b3.[;
:: TOPGEN_5:th 12
theorem
for b1, b2 being set holds
product <*b1,b2*>,[:b1,b2:] are_equipotent &
Card product <*b1,b2*> = (Card b1) *` Card b2;
:: TOPGEN_5:sch 1
scheme TOPGEN_5:sch 1
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> set,
F5 -> set}:
ex b1 being Function-like quasi_total Relation of F3(),F2() st
for b2 being Element of F1()
st b2 in F3()
holds (P1[b2] implies b1 . b2 = F4(b2)) & (P1[b2] or b1 . b2 = F5(b2))
provided
F3() c= F1()
and
for b1 being Element of F1()
st b1 in F3()
holds (P1[b1] implies F4(b1) in F2()) & (P1[b1] or F5(b1) in F2());
:: TOPGEN_5:sch 2
scheme TOPGEN_5:sch 2
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> set,
F5 -> set,
F6 -> set}:
ex b1 being Function-like quasi_total Relation of F3(),F2() st
for b2 being Element of F1()
st b2 in F3()
holds (P1[b2] implies b1 . b2 = F4(b2)) & (not (P1[b2]) & P2[b2] implies b1 . b2 = F5(b2)) & (not (P1[b2]) & not (P2[b2]) implies b1 . b2 = F6(b2))
provided
F3() c= F1()
and
for b1 being Element of F1()
st b1 in F3()
holds (P1[b1] implies F4(b1) in F2()) & (not (P1[b1]) & P2[b1] implies F5(b1) in F2()) & (not (P1[b1]) & not (P2[b1]) implies F6(b1) in F2());
:: TOPGEN_5:th 13
theorem
for b1, b2 being real set holds
|.|[b1,b2]|.| ^2 = b1 ^2 + (b2 ^2);
:: TOPGEN_5:th 14
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3, b4 being closed Element of bool the carrier of b1
for b5 being Function-like quasi_total continuous Relation of the carrier of b1 | b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b1 | b4,the carrier of b2
st b5 tolerates b6
holds b5 +* b6 is Function-like quasi_total continuous Relation of the carrier of b1 | (b3 \/ b4),the carrier of b2;
:: TOPGEN_5:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3, b4 being closed Element of bool the carrier of b1
st b3 misses b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b1 | b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b1 | b4,the carrier of b2 holds
b5 +* b6 is Function-like quasi_total continuous Relation of the carrier of b1 | (b3 \/ b4),the carrier of b2;
:: TOPGEN_5:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being open closed Element of bool the carrier of b1
for b4 being Function-like quasi_total continuous Relation of the carrier of b1 | b3,the carrier of b2
for b5 being Function-like quasi_total continuous Relation of the carrier of b1 | (b3 `),the carrier of b2 holds
b4 +* b5 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;
:: TOPGEN_5:th 17
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being real positive set holds
b2 in Ball(b2,b3);
:: TOPGEN_5:funcnot 1 => TOPGEN_5:func 1
definition
func y=0-line -> Element of bool the carrier of TOP-REAL 2 equals
{|[b1,0]| where b1 is Element of REAL: TRUE};
end;
:: TOPGEN_5:def 1
theorem
y=0-line = {|[b1,0]| where b1 is Element of REAL: TRUE};
:: TOPGEN_5:funcnot 2 => TOPGEN_5:func 2
definition
func y>=0-plane -> Element of bool the carrier of TOP-REAL 2 equals
{|[b1,b2]| where b1 is Element of REAL, b2 is Element of REAL: 0 <= b2};
end;
:: TOPGEN_5:def 2
theorem
y>=0-plane = {|[b1,b2]| where b1 is Element of REAL, b2 is Element of REAL: 0 <= b2};
:: TOPGEN_5:th 18
theorem
for b1, b2 being set holds
<*b1,b2*> in y=0-line
iff
b1 in REAL & b2 = 0;
:: TOPGEN_5:th 19
theorem
for b1, b2 being real set holds
|[b1,b2]| in y=0-line
iff
b2 = 0;
:: TOPGEN_5:th 20
theorem
Card y=0-line = continuum;
:: TOPGEN_5:th 21
theorem
for b1, b2 being set holds
<*b1,b2*> in y>=0-plane
iff
b1 in REAL &
(ex b3 being Element of REAL st
b2 = b3 & 0 <= b3);
:: TOPGEN_5:th 22
theorem
for b1, b2 being real set holds
|[b1,b2]| in y>=0-plane
iff
0 <= b2;
:: TOPGEN_5:funcreg 1
registration
cluster y=0-line -> non empty;
end;
:: TOPGEN_5:funcreg 2
registration
cluster y>=0-plane -> non empty;
end;
:: TOPGEN_5:th 23
theorem
y=0-line c= y>=0-plane;
:: TOPGEN_5:th 24
theorem
for b1, b2, b3 being real set
st 0 < b3
holds Ball(|[b1,b2]|,b3) c= y>=0-plane
iff
b3 <= b2;
:: TOPGEN_5:th 25
theorem
for b1, b2, b3 being real set
st 0 < b3 & 0 <= b2
holds Ball(|[b1,b2]|,b3) misses y=0-line
iff
b3 <= b2;
:: TOPGEN_5:th 26
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being real positive set
st |.b2 - b3.| <= b4 - b5
holds Ball(b3,b5) c= Ball(b2,b4);
:: TOPGEN_5:th 27
theorem
for b1 being real set
for b2, b3 being real positive set
st b2 <= b3
holds Ball(|[b1,b2]|,b2) c= Ball(|[b1,b3]|,b3);
:: TOPGEN_5:th 28
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Neighborhood_System of b1
for b4 being Neighborhood_System of b2
st b3 = b4
holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);
:: TOPGEN_5:funcnot 3 => TOPGEN_5:func 3
definition
func Niemytzki-plane -> non empty strict TopSpace-like TopStruct means
the carrier of it = y>=0-plane &
(ex b1 being Neighborhood_System of it st
(for b2 being Element of REAL holds
b1 . |[b2,0]| = {(Ball(|[b2,b3]|,b3)) \/ {|[b2,0]|} where b3 is Element of REAL: 0 < b3}) &
(for b2, b3 being Element of REAL
st 0 < b3
holds b1 . |[b2,b3]| = {(Ball(|[b2,b3]|,b4)) /\ y>=0-plane where b4 is Element of REAL: 0 < b4}));
end;
:: TOPGEN_5:def 3
theorem
for b1 being non empty strict TopSpace-like TopStruct holds
b1 = Niemytzki-plane
iff
the carrier of b1 = y>=0-plane &
(ex b2 being Neighborhood_System of b1 st
(for b3 being Element of REAL holds
b2 . |[b3,0]| = {(Ball(|[b3,b4]|,b4)) \/ {|[b3,0]|} where b4 is Element of REAL: 0 < b4}) &
(for b3, b4 being Element of REAL
st 0 < b4
holds b2 . |[b3,b4]| = {(Ball(|[b3,b4]|,b5)) /\ y>=0-plane where b5 is Element of REAL: 0 < b5}));
:: TOPGEN_5:th 29
theorem
y>=0-plane \ y=0-line is open Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 30
theorem
y=0-line is closed Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 31
theorem
for b1 being real set
for b2 being real positive set holds
(Ball(|[b1,b2]|,b2)) \/ {|[b1,0]|} is open Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 32
theorem
for b1 being real set
for b2, b3 being real positive set holds
(Ball(|[b1,b2]|,b3)) /\ y>=0-plane is open Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 33
theorem
for b1, b2 being real set
for b3 being real positive set
st b3 <= b2
holds Ball(|[b1,b2]|,b3) is open Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 34
theorem
for b1 being Element of the carrier of Niemytzki-plane
for b2 being real positive set holds
ex b3 being Element of the carrier of TOP-REAL 2 st
ex b4 being open Element of bool the carrier of Niemytzki-plane st
b1 in b4 &
b3 in b4 &
(for b5 being Element of the carrier of TOP-REAL 2
st b5 in b4
holds |.b5 - b3.| < b2);
:: TOPGEN_5:th 35
theorem
for b1, b2 being real set
for b3 being real positive set holds
ex b4, b5 being rational set st
|[b4,b5]| in Ball(|[b1,b2]|,b3) &
|[b4,b5]| <> |[b1,b2]|;
:: TOPGEN_5:th 36
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
st b1 = (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*>
for b2 being set holds
Cl (b1 \ {b2}) = [#] Niemytzki-plane;
:: TOPGEN_5:th 37
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
st b1 = y>=0-plane \ y=0-line
for b2 being set holds
Cl (b1 \ {b2}) = [#] Niemytzki-plane;
:: TOPGEN_5:th 38
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
st b1 = y>=0-plane \ y=0-line
holds Cl b1 = [#] Niemytzki-plane;
:: TOPGEN_5:th 39
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
st b1 = y=0-line
holds Cl b1 = b1 & Int b1 = {};
:: TOPGEN_5:th 40
theorem
(y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> is dense Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 41
theorem
(y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> is dense-in-itself Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 42
theorem
y>=0-plane \ y=0-line is dense Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 43
theorem
y>=0-plane \ y=0-line is dense-in-itself Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 44
theorem
y=0-line is nowhere_dense Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 45
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
st b1 = y=0-line
holds Der b1 is empty;
:: TOPGEN_5:th 46
theorem
for b1 being Element of bool y=0-line holds
b1 is closed Element of bool the carrier of Niemytzki-plane;
:: TOPGEN_5:th 47
theorem
RAT is dense Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_5:th 48
theorem
Sorgenfrey-line is separable;
:: TOPGEN_5:th 49
theorem
Niemytzki-plane is separable;
:: TOPGEN_5:th 50
theorem
Niemytzki-plane is being_T1;
:: TOPGEN_5:th 51
theorem
Niemytzki-plane is not being_T4;
:: TOPGEN_5:attrnot 1 => TOPGEN_5:attr 1
definition
let a1 be TopSpace-like TopStruct;
attr a1 is Tychonoff means
a1 is being_T1 &
(for b1 being closed Element of bool the carrier of a1
for b2 being Element of the carrier of a1
st b2 in b1 `
holds ex b3 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of I[01] st
b3 . b2 = 0 & b3 .: b1 c= {1});
end;
:: TOPGEN_5:dfs 4
definiens
let a1 be TopSpace-like TopStruct;
To prove
a1 is Tychonoff
it is sufficient to prove
thus a1 is being_T1 &
(for b1 being closed Element of bool the carrier of a1
for b2 being Element of the carrier of a1
st b2 in b1 `
holds ex b3 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of I[01] st
b3 . b2 = 0 & b3 .: b1 c= {1});
:: TOPGEN_5:def 4
theorem
for b1 being TopSpace-like TopStruct holds
b1 is Tychonoff
iff
b1 is being_T1 &
(for b2 being closed Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2 `
holds ex b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of I[01] st
b4 . b3 = 0 & b4 .: b2 c= {1});
:: TOPGEN_5:condreg 2
registration
cluster TopSpace-like Tychonoff -> being_T3 being_T1 (TopStruct);
end;
:: TOPGEN_5:condreg 3
registration
cluster non empty TopSpace-like being_T4 being_T1 -> Tychonoff (TopStruct);
end;
:: TOPGEN_5:th 52
theorem
for b1 being TopSpace-like being_T1 TopStruct
st b1 is Tychonoff
for b2 being prebasis of b1
for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b3 in b4 & b4 in b2
holds ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of I[01] st
b5 . b3 = 0 & b5 .: (b4 `) c= {1};
:: TOPGEN_5:condreg 4
registration
let a1 be set;
let a2 be non empty real-membered set;
cluster -> real-valued (Relation of a1,a2);
end;
:: TOPGEN_5:th 53
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty SubSpace of R^1
for b3, b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b5 being Element of bool the carrier of b1
st for b6 being Element of the carrier of b1 holds
b6 in b5
iff
b3 . b6 <= b4 . b6
holds b5 is closed(b1);
:: TOPGEN_5:th 54
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty SubSpace of R^1
for b3, b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
for b6 being Element of the carrier of b1 holds
b5 . b6 = max(b3 . b6,b4 . b6);
:: TOPGEN_5:th 55
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of R^1
for b3 being non empty finite set
for b4 being Function-yielding ManySortedSet of b3
st for b5 being set
st b5 in b3
holds b4 . b5 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
holds ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
for b6 being Element of the carrier of b1
for b7 being non empty finite Element of bool REAL
st b7 = proj2 ((commute b4) . b6)
holds b5 . b6 = upper_bound b7;
:: TOPGEN_5:th 56
theorem
for b1 being non empty TopSpace-like being_T1 TopStruct
for b2 being prebasis of b1
st for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b3 in b4 & b4 in b2
holds ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of I[01] st
b5 . b3 = 0 & b5 .: (b4 `) c= {1}
holds b1 is Tychonoff;
:: TOPGEN_5:th 57
theorem
Sorgenfrey-line is being_T1;
:: TOPGEN_5:th 58
theorem
for b1 being real set holds
halfline b1 is closed Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_5:th 59
theorem
for b1 being real set holds
left_closed_halfline b1 is closed Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_5:th 60
theorem
for b1 being real set holds
right_closed_halfline b1 is closed Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_5:th 61
theorem
for b1, b2 being real set holds
[.b1,b2.[ is closed Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_5:th 62
theorem
for b1 being real set
for b2 being rational set
st b1 < b2
holds ex b3 being Function-like quasi_total continuous Relation of the carrier of Sorgenfrey-line,the carrier of I[01] st
for b4 being Element of the carrier of Sorgenfrey-line holds
(b4 in [.b1,b2.[ implies b3 . b4 = 0) &
(b4 in [.b1,b2.[ or b3 . b4 = 1);
:: TOPGEN_5:th 63
theorem
Sorgenfrey-line is Tychonoff;
:: TOPGEN_5:funcnot 4 => TOPGEN_5:func 4
definition
let a1 be real set;
let a2 be real positive set;
func +(A1,A2) -> Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] means
it . |[a1,0]| = 0 &
(for b1 being real set
for b2 being real non negative set holds
((b1 = a1 implies b2 <> 0) &
not |[b1,b2]| in Ball(|[a1,a2]|,a2) implies it . |[b1,b2]| = 1) &
(|[b1,b2]| in Ball(|[a1,a2]|,a2) implies it . |[b1,b2]| = |.|[a1,0]| - |[b1,b2]|.| ^2 / ((2 * a2) * b2)));
end;
:: TOPGEN_5:def 5
theorem
for b1 being real set
for b2 being real positive set
for b3 being Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] holds
b3 = +(b1,b2)
iff
b3 . |[b1,0]| = 0 &
(for b4 being real set
for b5 being real non negative set holds
((b4 = b1 implies b5 <> 0) &
not |[b4,b5]| in Ball(|[b1,b2]|,b2) implies b3 . |[b4,b5]| = 1) &
(|[b4,b5]| in Ball(|[b1,b2]|,b2) implies b3 . |[b4,b5]| = |.|[b1,0]| - |[b4,b5]|.| ^2 / ((2 * b2) * b5)));
:: TOPGEN_5:th 64
theorem
for b1 being Element of the carrier of TOP-REAL 2
st 0 <= b1 `2
for b2 being real set
for b3 being real positive set
st (+(b2,b3)) . b1 = 0
holds b1 = |[b2,0]|;
:: TOPGEN_5:th 65
theorem
for b1, b2 being real set
for b3 being real positive set
st b1 <> b2
holds (+(b1,b3)) . |[b2,0]| = 1;
:: TOPGEN_5:th 66
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being real set
for b3, b4 being real positive set
st b3 <= 1 &
|.b1 - |[b2,b4 * b3]|.| = b4 * b3 &
b1 `2 <> 0
holds (+(b2,b4)) . b1 = b3;
:: TOPGEN_5:th 67
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being real set
for b4 being real positive set
st 0 <= b3 &
b3 <= 1 &
|.b1 - |[b2,b4 * b3]|.| < b4 * b3
holds (+(b2,b4)) . b1 < b3;
:: TOPGEN_5:th 68
theorem
for b1 being Element of the carrier of TOP-REAL 2
st 0 <= b1 `2
for b2, b3 being real set
for b4 being real positive set
st 0 <= b3 &
b3 < 1 &
b4 * b3 < |.b1 - |[b2,b4 * b3]|.|
holds b3 < (+(b2,b4)) . b1;
:: TOPGEN_5:th 69
theorem
for b1 being Element of the carrier of TOP-REAL 2
st 0 <= b1 `2
for b2, b3, b4 being real set
for b5 being real positive set
st 0 <= b3 & b4 <= 1 & (+(b2,b5)) . b1 in ].b3,b4.[
holds ex b6 being real positive set st
b6 <= b1 `2 &
Ball(b1,b6) c= (+(b2,b5)) " ].b3,b4.[;
:: TOPGEN_5:th 70
theorem
for b1 being real set
for b2, b3 being real positive set holds
Ball(|[b1,b3 * b2]|,b3 * b2) c= (+(b1,b3)) " ].0,b2.[;
:: TOPGEN_5:th 71
theorem
for b1 being real set
for b2, b3 being real positive set holds
(Ball(|[b1,b3 * b2]|,b3 * b2)) \/ {|[b1,0]|} c= (+(b1,b3)) " [.0,b2.[;
:: TOPGEN_5:th 72
theorem
for b1 being Element of the carrier of TOP-REAL 2
st 0 <= b1 `2
for b2, b3 being real set
for b4 being real positive set
st 0 < (+(b2,b4)) . b1 & (+(b2,b4)) . b1 < b3 & b3 <= 1
holds b1 in Ball(|[b2,b4 * b3]|,b4 * b3);
:: TOPGEN_5:th 73
theorem
for b1 being Element of the carrier of TOP-REAL 2
st 0 < b1 `2
for b2, b3 being real set
for b4 being real positive set
st 0 <= b3 & b3 < (+(b2,b4)) . b1
holds ex b5 being real positive set st
b5 <= b1 `2 &
Ball(b1,b5) c= (+(b2,b4)) " ].b3,1.];
:: TOPGEN_5:th 74
theorem
for b1 being Element of the carrier of TOP-REAL 2
st b1 `2 = 0
for b2 being real set
for b3 being real positive set
st (+(b2,b3)) . b1 = 1
holds ex b4 being real positive set st
(Ball(|[b1 `1,b4]|,b4)) \/ {b1} c= (+(b2,b3)) " {1};
:: TOPGEN_5:th 75
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Basis of b1 holds
{b4 /\ [#] b2 where b4 is Element of bool the carrier of b1: b4 in b3 & b4 meets [#] b2} is Basis of b2;
:: TOPGEN_5:th 76
theorem
{].b1,b2.[ where b1 is Element of REAL, b2 is Element of REAL: b1 < b2} is Basis of R^1;
:: TOPGEN_5:th 77
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being set
st b2 in b4 &
b3 in b4 &
b4 \/ {b2 \/ b3} is Basis of b1
holds b4 is Basis of b1;
:: TOPGEN_5:th 78
theorem
({[.0,b1.[ where b1 is Element of REAL: 0 < b1 & b1 <= 1} \/ {].b1,1.] where b1 is Element of REAL: 0 <= b1 & b1 < 1}) \/ {].b1,b2.[ where b1 is Element of REAL, b2 is Element of REAL: 0 <= b1 & b1 < b2 & b2 <= 1} is Basis of I[01];
:: TOPGEN_5:th 79
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of I[01] holds
b2 is continuous(b1, I[01])
iff
for b3, b4 being real set
st 0 <= b3 & b3 < 1 & 0 < b4 & b4 <= 1
holds b2 " [.0,b4.[ is open(b1) & b2 " ].b3,1.] is open(b1);
:: TOPGEN_5:funcreg 3
registration
let a1 be real set;
let a2 be real positive set;
cluster +(a1,a2) -> Function-like quasi_total continuous;
end;
:: TOPGEN_5:th 80
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
for b2 being Element of REAL
for b3 being real positive set
st b1 = (Ball(|[b2,b3]|,b3)) \/ {|[b2,0]|}
holds ex b4 being Function-like quasi_total continuous Relation of the carrier of Niemytzki-plane,the carrier of I[01] st
b4 . |[b2,0]| = 0 &
(for b5, b6 being real set holds
(|[b5,b6]| in b1 ` implies b4 . |[b5,b6]| = 1) &
(|[b5,b6]| in b1 \ {|[b2,0]|} implies b4 . |[b5,b6]| = |.|[b2,0]| - |[b5,b6]|.| ^2 / ((2 * b3) * b6)));
:: TOPGEN_5:funcnot 5 => TOPGEN_5:func 5
definition
let a1, a2 be real set;
let a3 be real positive set;
func +(A1,A2,A3) -> Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] means
for b1 being real set
for b2 being real non negative set holds
(|[b1,b2]| in Ball(|[a1,a2]|,a3) or it . |[b1,b2]| = 1) &
(|[b1,b2]| in Ball(|[a1,a2]|,a3) implies it . |[b1,b2]| = |.|[a1,a2]| - |[b1,b2]|.| / a3);
end;
:: TOPGEN_5:def 6
theorem
for b1, b2 being real set
for b3 being real positive set
for b4 being Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] holds
b4 = +(b1,b2,b3)
iff
for b5 being real set
for b6 being real non negative set holds
(|[b5,b6]| in Ball(|[b1,b2]|,b3) or b4 . |[b5,b6]| = 1) &
(|[b5,b6]| in Ball(|[b1,b2]|,b3) implies b4 . |[b5,b6]| = |.|[b1,b2]| - |[b5,b6]|.| / b3);
:: TOPGEN_5:th 81
theorem
for b1 being Element of the carrier of TOP-REAL 2
st 0 <= b1 `2
for b2 being real set
for b3 being real non negative set
for b4 being real positive set holds
(+(b2,b3,b4)) . b1 = 0
iff
b1 = |[b2,b3]|;
:: TOPGEN_5:th 82
theorem
for b1 being real set
for b2 being real non negative set
for b3, b4 being real positive set
st b4 <= 1
holds (+(b1,b2,b3)) " [.0,b4.[ = (Ball(|[b1,b2]|,b3 * b4)) /\ y>=0-plane;
:: TOPGEN_5:th 83
theorem
for b1 being Element of the carrier of TOP-REAL 2
st 0 < b1 `2
for b2 being real set
for b3 being real non negative set
for b4, b5 being real positive set
st b3 < (+(b2,b4,b5)) . b1
holds b5 * b3 < |.|[b2,b4]| - b1.| &
(Ball(b1,|.|[b2,b4]| - b1.| - (b5 * b3))) /\ y>=0-plane c= (+(b2,b4,b5)) " ].b3,1.];
:: TOPGEN_5:th 84
theorem
for b1 being Element of the carrier of TOP-REAL 2
st b1 `2 = 0
for b2 being real set
for b3 being real non negative set
for b4, b5 being real positive set
st b3 < (+(b2,b4,b5)) . b1
holds b5 * b3 < |.|[b2,b4]| - b1.| &
(ex b6 being real positive set st
b6 = (|.|[b2,b4]| - b1.| - (b5 * b3)) / 2 &
(Ball(|[b1 `1,b6]|,b6)) \/ {b1} c= (+(b2,b4,b5)) " ].b3,1.]);
:: TOPGEN_5:funcreg 4
registration
let a1 be real set;
let a2, a3 be real positive set;
cluster +(a1,a2,a3) -> Function-like quasi_total continuous;
end;
:: TOPGEN_5:th 85
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
for b2, b3 being Element of REAL
for b4 being real positive set
st 0 < b3 &
b1 = (Ball(|[b2,b3]|,b4)) /\ y>=0-plane
holds ex b5 being Function-like quasi_total continuous Relation of the carrier of Niemytzki-plane,the carrier of I[01] st
b5 . |[b2,b3]| = 0 &
(for b6, b7 being real set holds
(|[b6,b7]| in b1 ` implies b5 . |[b6,b7]| = 1) &
(|[b6,b7]| in b1 implies b5 . |[b6,b7]| = |.|[b2,b3]| - |[b6,b7]|.| / b4));
:: TOPGEN_5:th 86
theorem
Niemytzki-plane is being_T1;
:: TOPGEN_5:th 87
theorem
Niemytzki-plane is Tychonoff;