:: NAGATA_1 semantic presentation
Lemma29:
for r, s, t being Real st r >= 0 & s >= 0 & r + s < t holds
( r < t & s < t )
Lemma34:
for r1, r2, r3, r4 being Real holds abs (r1 - r4) <= ((abs (r1 - r2)) + (abs (r2 - r3))) + (abs (r3 - r4))
:: 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
Lemma81:
for T being non empty TopSpace
for O being open Subset of T
for A being Subset of T st O meets Cl A holds
O meets A
Lemma82:
for T being non empty TopSpace
for F being Subset-Family of T
for A being Subset of T st A in F holds
Cl A c= Cl (union F)
theorem Th9: :: NAGATA_1:9
theorem Th10: :: NAGATA_1:10
theorem Th11: :: NAGATA_1:11
:: deftheorem Def2 defines sigma_discrete NAGATA_1:def 2 :
Lemma93:
for T being non empty TopSpace ex Un being FamilySequence of T st Un 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
Lemma149:
for T being non empty TopSpace
for U being open Subset of T
for A being Subset of T st A is closed & U is open holds
U \ A is open
theorem Th17: :: NAGATA_1:17
theorem Th18: :: NAGATA_1:18
theorem Th19: :: NAGATA_1:19
theorem Th20: :: NAGATA_1:20
Lemma188:
for r being Real
for A being Point of RealSpace st r > 0 holds
A in Ball A,r
:: 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 X be
set ;
let r be
Real;
let f be
Function of
X,
REAL ;
deffunc H1(
Element of
X)
-> set =
min r,
(f . a1);
func min c2,
c3 -> Function of
a1,
REAL means :
Def9:
:: NAGATA_1:def 9
for
x being
set st
x in X holds
it . x = min r,
(f . x);
existence
ex b1 being Function of X, REAL st
for x being set st x in X holds
b1 . x = min r,(f . x)
uniqueness
for b1, b2 being Function of X, REAL st ( for x being set st x in X holds
b1 . x = min r,(f . x) ) & ( for x being set st x in X holds
b2 . x = min r,(f . x) ) holds
b1 = b2
end;
:: deftheorem Def9 defines min NAGATA_1:def 9 :
theorem Th27: :: NAGATA_1:27
:: deftheorem Def10 defines is_a_pseudometric_of NAGATA_1:def 10 :
Lemma262:
for X being set
for f being Function of [:X,X:], REAL holds
( f is_a_pseudometric_of X iff for a, b, c being Element of X holds
( f . a,a = 0 & f . a,b = f . b,a & f . a,c <= (f . a,b) + (f . b,c) ) )
theorem Th28: :: NAGATA_1:28
Lemma265:
for r being Real
for X being non empty set
for f being Function of [:X,X:], REAL
for x, y being Element of X holds (min r,f) . x,y = min r,(f . x,y)
theorem Th29: :: NAGATA_1:29
theorem Th30: :: NAGATA_1:30
theorem Th31: :: NAGATA_1:31