:: NAGATA_1 semantic presentation
Lemma1:
for b1, b2, b3 being Real holds
( b1 >= 0 & b2 >= 0 & b1 + b2 < b3 implies ( b1 < b3 & b2 < b3 ) )
Lemma2:
for b1, b2, b3, b4 being Real holds abs (b1 - b4) <= ((abs (b1 - b2)) + (abs (b2 - b3))) + (abs (b3 - b4))
:: deftheorem Def1 defines discrete NAGATA_1:def 1 :
theorem Th1: :: NAGATA_1:1
theorem Th2: :: NAGATA_1:2
theorem Th3: :: NAGATA_1:3
theorem Th4: :: NAGATA_1:4
theorem Th5: :: NAGATA_1:5
theorem Th6: :: NAGATA_1:6
theorem Th7: :: NAGATA_1:7
theorem Th8: :: NAGATA_1:8
Lemma8:
for b1 being non empty TopSpace
for b2 being open Subset of b1
for b3 being Subset of b1 holds
not ( b2 meets Cl b3 & not b2 meets b3 )
Lemma9:
for b1 being non empty TopSpace
for b2 being Subset-Family of b1
for b3 being Subset of b1 holds
( b3 in b2 implies Cl b3 c= Cl (union b2) )
theorem Th9: :: NAGATA_1:9
theorem Th10: :: NAGATA_1:10
theorem Th11: :: NAGATA_1:11
:: deftheorem Def2 defines sigma_discrete NAGATA_1:def 2 :
Lemma12:
for b1 being non empty TopSpace holds
ex b2 being FamilySequence of b1 st b2 is sigma_discrete
:: deftheorem Def3 defines sigma_locally_finite NAGATA_1:def 3 :
:: deftheorem Def4 defines sigma_discrete NAGATA_1:def 4 :
theorem Th12: :: NAGATA_1:12
theorem Th13: :: NAGATA_1:13
:: deftheorem Def5 defines Basis_sigma_discrete NAGATA_1:def 5 :
:: deftheorem Def6 defines Basis_sigma_locally_finite NAGATA_1:def 6 :
theorem Th14: :: NAGATA_1:14
theorem Th15: :: NAGATA_1:15
theorem Th16: :: NAGATA_1:16
Lemma17:
for b1 being non empty TopSpace
for b2 being open Subset of b1
for b3 being Subset of b1 holds
( b3 is closed & b2 is open implies b2 \ b3 is open )
theorem Th17: :: NAGATA_1:17
theorem Th18: :: NAGATA_1:18
theorem Th19: :: NAGATA_1:19
theorem Th20: :: NAGATA_1:20
Lemma21:
for b1 being Real
for b2 being Point of RealSpace holds
( b1 > 0 implies b2 in Ball b2,b1 )
:: deftheorem Def7 defines + NAGATA_1:def 7 :
theorem Th21: :: NAGATA_1:21
theorem Th22: :: NAGATA_1:22
theorem Th23: :: NAGATA_1:23
theorem Th24: :: NAGATA_1:24
:: deftheorem Def8 defines Toler NAGATA_1:def 8 :
theorem Th25: :: NAGATA_1:25
theorem Th26: :: NAGATA_1:26
definition
let c
1 be
set ;
let c
2 be
Real;
let c
3 be
Function of c
1,
REAL ;
deffunc H
1(
Element of c
1)
-> set =
min c
2,
(c3 . a1);
func min c
2,c
3 -> Function of a
1,
REAL means :
Def9:
:: NAGATA_1:def 9
for b
1 being
set holds
( b
1 in a
1 implies a
4 . b
1 = min a
2,
(a3 . b1) );
existence
ex b1 being Function of c1, REAL st
for b2 being set holds
( b2 in c1 implies b1 . b2 = min c2,(c3 . b2) )
uniqueness
for b1, b2 being Function of c1, REAL holds
( ( for b3 being set holds
( b3 in c1 implies b1 . b3 = min c2,(c3 . b3) ) ) & ( for b3 being set holds
( b3 in c1 implies b2 . b3 = min c2,(c3 . b3) ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines min NAGATA_1:def 9 :
for b
1 being
set for b
2 being
Realfor b
3, b
4 being
Function of b
1,
REAL holds
( b
4 = min b
2,b
3 iff for b
5 being
set holds
( b
5 in b
1 implies b
4 . b
5 = min b
2,
(b3 . b5) ) );
theorem Th27: :: NAGATA_1:27
:: deftheorem Def10 defines is_a_pseudometric_of NAGATA_1:def 10 :
Lemma29:
for b1 being set
for b2 being Function of [:b1,b1:], REAL holds
( b2 is_a_pseudometric_of b1 iff for b3, b4, b5 being Element of b1 holds
( b2 . b3,b3 = 0 & b2 . b3,b4 = b2 . b4,b3 & b2 . b3,b5 <= (b2 . b3,b4) + (b2 . b4,b5) ) )
theorem Th28: :: NAGATA_1:28
Lemma31:
for b1 being Real
for b2 being non empty set
for b3 being Function of [:b2,b2:], REAL
for b4, b5 being Element of b2 holds (min b1,b3) . b4,b5 = min b1,(b3 . b4,b5)
theorem Th29: :: NAGATA_1:29
theorem Th30: :: NAGATA_1:30
theorem Th31: :: NAGATA_1:31