:: TSP_1 semantic presentation
:: deftheorem Def1 defines SubSpace TSP_1:def 1 :
theorem Th1: :: TSP_1:1
canceled;
theorem Th2: :: TSP_1:2
:: deftheorem Def2 defines SubSpace TSP_1:def 2 :
theorem Th3: :: TSP_1:3
canceled;
theorem Th4: :: TSP_1:4
:: deftheorem Def3 defines T_0 TSP_1:def 3 :
for b
1 being
TopStruct holds
( b
1 is
T_0 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 ) ) & ( for b
4 being
Subset of b
1 holds
not ( b
4 is
open & not b
2 in b
4 & b
3 in b
4 ) ) ) ) );
:: deftheorem Def4 defines T_0 TSP_1:def 4 :
for b
1 being
TopStruct holds
( b
1 is
T_0 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
closed & b
2 in b
4 & not b
3 in b
4 ) ) & ( for b
4 being
Subset of b
1 holds
not ( b
4 is
closed & not b
2 in b
4 & b
3 in b
4 ) ) ) ) );
Lemma7:
for b1 being non empty non trivial anti-discrete TopStruct holds
not b1 is T_0
Lemma8:
for b1 being non empty TopSpace
for b2 being Point of b1 holds b2 in Cl {b2}
Lemma9:
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b3 c= Cl b2 implies Cl b3 c= Cl b2 )
by TOPS_1:31;
:: deftheorem Def5 defines T_0 TSP_1:def 5 :
:: deftheorem Def6 defines T_0 TSP_1:def 6 :
:: deftheorem Def7 defines T_0 TSP_1:def 7 :
:: deftheorem Def8 defines T_0 TSP_1:def 8 :
for b
1 being
TopStruct for b
2 being
Subset of b
1 holds
( b
2 is
T_0 iff for b
3, b
4 being
Point of b
1 holds
not ( b
3 in b
2 & b
4 in b
2 & b
3 <> b
4 & ( for b
5 being
Subset of b
1 holds
not ( b
5 is
open & b
3 in b
5 & not b
4 in b
5 ) ) & ( for b
5 being
Subset of b
1 holds
not ( b
5 is
open & not b
3 in b
5 & b
4 in b
5 ) ) ) );
:: deftheorem Def9 defines T_0 TSP_1:def 9 :
theorem Th5: :: TSP_1:5
theorem Th6: :: TSP_1:6
theorem Th7: :: TSP_1:7
theorem Th8: :: TSP_1:8
theorem Th9: :: TSP_1:9
theorem Th10: :: TSP_1:10
theorem Th11: :: TSP_1:11
theorem Th12: :: TSP_1:12
:: deftheorem Def10 defines T_0 TSP_1:def 10 :
:: deftheorem Def11 defines T_0 TSP_1:def 11 :
:: deftheorem Def12 defines T_0 TSP_1:def 12 :
theorem Th13: :: TSP_1:13
theorem Th14: :: TSP_1:14
:: deftheorem Def13 defines T_0 TSP_1:def 13 :
for b
1 being
TopStruct for b
2 being
SubSpace of b
1 holds
( b
2 is
T_0 iff ( b
2 is
empty or for b
3, b
4 being
Point of b
1 holds
not ( b
3 is
Point of b
2 & b
4 is
Point of b
2 & b
3 <> b
4 & ( for b
5 being
Subset of b
1 holds
not ( b
5 is
open & b
3 in b
5 & not b
4 in b
5 ) ) & ( for b
5 being
Subset of b
1 holds
not ( b
5 is
open & not b
3 in b
5 & b
4 in b
5 ) ) ) ) );
:: deftheorem Def14 defines T_0 TSP_1:def 14 :
theorem Th15: :: TSP_1:15
theorem Th16: :: TSP_1:16
theorem Th17: :: TSP_1:17
theorem Th18: :: TSP_1:18
theorem Th19: :: TSP_1:19
theorem Th20: :: TSP_1:20
theorem Th21: :: TSP_1:21