:: T_0TOPSP semantic presentation
theorem Th1: :: T_0TOPSP:1
theorem Th2: :: T_0TOPSP:2
for b
1, b
2 being non
empty set for b
3 being
Function of b
1,b
2for b
4 being
Subset of b
1 holds
( ( for b
5, b
6 being
Element of b
1 holds
( b
5 in b
4 & b
3 . b
5 = b
3 . b
6 implies b
6 in b
4 ) ) implies b
3 " (b3 .: b4) = b
4 )
:: deftheorem Def1 defines are_homeomorphic T_0TOPSP:def 1 :
:: deftheorem Def2 defines open T_0TOPSP:def 2 :
:: deftheorem Def3 defines Indiscernibility T_0TOPSP:def 3 :
:: deftheorem Def4 defines Indiscernible T_0TOPSP:def 4 :
:: deftheorem Def5 defines T_0-reflex T_0TOPSP:def 5 :
:: deftheorem Def6 defines T_0-canonical_map T_0TOPSP:def 6 :
theorem Th3: :: T_0TOPSP:3
theorem Th4: :: T_0TOPSP:4
theorem Th5: :: T_0TOPSP:5
theorem Th6: :: T_0TOPSP:6
theorem Th7: :: T_0TOPSP:7
theorem Th8: :: T_0TOPSP:8
theorem Th9: :: T_0TOPSP:9
theorem Th10: :: T_0TOPSP:10
theorem Th11: :: T_0TOPSP:11
theorem Th12: :: T_0TOPSP:12
Lemma15:
for b1 being non empty TopSpace
for b2, b3 being Point of (T_0-reflex b1) holds
not ( b2 <> b3 & ( for b4 being Subset of (T_0-reflex b1) holds
not ( b4 is open & ( ( b2 in b4 & not b3 in b4 ) or ( b3 in b4 & not b2 in b4 ) ) ) ) )
:: deftheorem Def7 defines discerning T_0TOPSP:def 7 :
for b
1 being
TopStruct holds
( b
1 is
discerning iff ( b
1 is
empty or for b
2, b
3 being
Point of b
1 holds
not ( b
2 <> b
3 & ( for b
4 being
Subset of b
1 holds
not ( b
4 is
open & ( ( b
2 in b
4 & not b
3 in b
4 ) or ( b
3 in b
4 & not b
2 in b
4 ) ) ) ) ) ) );
theorem Th13: :: T_0TOPSP:13
theorem Th14: :: T_0TOPSP:14
theorem Th15: :: T_0TOPSP:15
theorem Th16: :: T_0TOPSP:16
theorem Th17: :: T_0TOPSP:17