:: URYSOHN1 semantic presentation
theorem Th1: :: URYSOHN1:1
theorem Th2: :: URYSOHN1:2
definition
canceled;
canceled;
end;
:: deftheorem Def1 URYSOHN1:def 1 :
canceled;
:: deftheorem Def2 URYSOHN1:def 2 :
canceled;
:: deftheorem Def3 defines dyadic URYSOHN1:def 3 :
:: deftheorem Def4 defines DYADIC URYSOHN1:def 4 :
:: deftheorem Def5 defines DOM URYSOHN1:def 5 :
theorem Th3: :: URYSOHN1:3
canceled;
theorem Th4: :: URYSOHN1:4
canceled;
theorem Th5: :: URYSOHN1:5
theorem Th6: :: URYSOHN1:6
theorem Th7: :: URYSOHN1:7
theorem Th8: :: URYSOHN1:8
canceled;
theorem Th9: :: URYSOHN1:9
:: deftheorem Def6 defines dyad URYSOHN1:def 6 :
theorem Th10: :: URYSOHN1:10
theorem Th11: :: URYSOHN1:11
theorem Th12: :: URYSOHN1:12
theorem Th13: :: URYSOHN1:13
theorem Th14: :: URYSOHN1:14
theorem Th15: :: URYSOHN1:15
:: deftheorem Def7 defines axis URYSOHN1:def 7 :
theorem Th16: :: URYSOHN1:16
theorem Th17: :: URYSOHN1:17
theorem Th18: :: URYSOHN1:18
theorem Th19: :: URYSOHN1:19
canceled;
theorem Th20: :: URYSOHN1:20
theorem Th21: :: URYSOHN1:21
theorem Th22: :: URYSOHN1:22
theorem Th23: :: URYSOHN1:23
:: deftheorem Def8 defines Nbhd URYSOHN1:def 8 :
theorem Th24: :: URYSOHN1:24
for b
1 being non
empty TopSpacefor b
2 being
Subset of b
1 holds
( b
2 is
open iff for b
3 being
Point of b
1 holds
not ( b
3 in b
2 & ( for b
4 being
Nbhd of b
3,b
1 holds
not b
4 c= b
2 ) ) )
theorem Th25: :: URYSOHN1:25
canceled;
theorem Th26: :: URYSOHN1:26
:: deftheorem Def9 defines being_T1 URYSOHN1:def 9 :
for b
1 being
TopStruct holds
( b
1 is
being_T1 iff for b
2, b
3 being
Point of b
1 holds
not ( not b
2 = b
3 & ( for b
4, b
5 being
Subset of b
1 holds
not ( b
4 is
open & b
5 is
open & b
2 in b
4 & not b
3 in b
4 & b
3 in b
5 & not b
2 in b
5 ) ) ) );
theorem Th27: :: URYSOHN1:27
theorem Th28: :: URYSOHN1:28
theorem Th29: :: URYSOHN1:29
theorem Th30: :: URYSOHN1:30
theorem Th31: :: URYSOHN1:31
:: deftheorem Def10 defines Between URYSOHN1:def 10 :
theorem Th32: :: URYSOHN1:32