:: YELLOW19 semantic presentation
theorem Th1: :: YELLOW19:1
canceled;
theorem Th2: :: YELLOW19:2
:: deftheorem Def1 defines NeighborhoodSystem YELLOW19:def 1 :
theorem Th3: :: YELLOW19:3
theorem Th4: :: YELLOW19:4
theorem Th5: :: YELLOW19:5
theorem Th6: :: YELLOW19:6
:: deftheorem Def2 defines Subset YELLOW19:def 2 :
theorem Th7: :: YELLOW19:7
theorem Th8: :: YELLOW19:8
theorem Th9: :: YELLOW19:9
theorem Th10: :: YELLOW19:10
:: deftheorem Def3 defines a_filter YELLOW19:def 3 :
theorem Th11: :: YELLOW19:11
theorem Th12: :: YELLOW19:12
theorem Th13: :: YELLOW19:13
definition
let c
1 be non
empty 1-sorted ;
let c
2 be non
empty Subset of c
1;
let c
3 be
Filter of
(BoolePoset c2);
func a_net c
3 -> non
empty strict NetStr of a
1 means :
Def4:
:: YELLOW19:def 4
( the
carrier of a
4 = { [b1,b2] where B is Element of a1, B is Element of a3 : b1 in b2 } & ( for b
1, b
2 being
Element of a
4 holds
( b
1 <= b
2 iff b
2 `2 c= b
1 `2 ) ) & ( for b
1 being
Element of a
4 holds a
4 . b
1 = b
1 `1 ) );
existence
ex b1 being non empty strict NetStr of c1 st
( the carrier of b1 = { [b2,b3] where B is Element of c1, B is Element of c3 : b2 in b3 } & ( for b2, b3 being Element of b1 holds
( b2 <= b3 iff b3 `2 c= b2 `2 ) ) & ( for b2 being Element of b1 holds b1 . b2 = b2 `1 ) )
uniqueness
for b1, b2 being non empty strict NetStr of c1 holds
( the carrier of b1 = { [b3,b4] where B is Element of c1, B is Element of c3 : b3 in b4 } & ( for b3, b4 being Element of b1 holds
( b3 <= b4 iff b4 `2 c= b3 `2 ) ) & ( for b3 being Element of b1 holds b1 . b3 = b3 `1 ) & the carrier of b2 = { [b3,b4] where B is Element of c1, B is Element of c3 : b3 in b4 } & ( for b3, b4 being Element of b2 holds
( b3 <= b4 iff b4 `2 c= b3 `2 ) ) & ( for b3 being Element of b2 holds b2 . b3 = b3 `1 ) implies b1 = b2 )
end;
:: deftheorem Def4 defines a_net YELLOW19:def 4 :
theorem Th14: :: YELLOW19:14
theorem Th15: :: YELLOW19:15
theorem Th16: :: YELLOW19:16
theorem Th17: :: YELLOW19:17
theorem Th18: :: YELLOW19:18
theorem Th19: :: YELLOW19:19
canceled;
theorem Th20: :: YELLOW19:20
theorem Th21: :: YELLOW19:21
theorem Th22: :: YELLOW19:22
theorem Th23: :: YELLOW19:23
theorem Th24: :: YELLOW19:24
theorem Th25: :: YELLOW19:25
theorem Th26: :: YELLOW19:26
theorem Th27: :: YELLOW19:27
theorem Th28: :: YELLOW19:28
theorem Th29: :: YELLOW19:29
theorem Th30: :: YELLOW19:30
theorem Th31: :: YELLOW19:31
theorem Th32: :: YELLOW19:32
Lemma24:
for b1 being non empty TopSpace holds
( b1 is compact implies for b2 being net of b1 holds
ex b3 being Point of b1 st b3 is_a_cluster_point_of b2 )
Lemma25:
for b1 being non empty TopSpace holds
( ( for b2 being net of b1 holds
not ( b2 in NetUniv b1 & ( for b3 being Point of b1 holds
not b3 is_a_cluster_point_of b2 ) ) ) implies b1 is compact )
theorem Th33: :: YELLOW19:33
theorem Th34: :: YELLOW19:34
theorem Th35: :: YELLOW19:35
theorem Th36: :: YELLOW19:36
theorem Th37: :: YELLOW19:37
:: deftheorem Def5 defines Cauchy YELLOW19:def 5 :
theorem Th38: :: YELLOW19:38