:: FINTOPO5 semantic presentation
theorem Th1: :: FINTOPO5:1
theorem Th2: :: FINTOPO5:2
for b
1 being
Nat holds
( not ( b
1 > 0 & not
Seg b
1 <> {} ) & not (
Seg b
1 <> {} & not b
1 > 0 ) )
:: deftheorem Def1 defines is_homeomorphism FINTOPO5:def 1 :
theorem Th3: :: FINTOPO5:3
theorem Th4: :: FINTOPO5:4
theorem Th5: :: FINTOPO5:5
theorem Th6: :: FINTOPO5:6
theorem Th7: :: FINTOPO5:7
theorem Th8: :: FINTOPO5:8
theorem Th9: :: FINTOPO5:9
theorem Th10: :: FINTOPO5:10
definition
let c
1, c
2 be
Nat;
func Nbdl2 c
1,c
2 -> Function of
[:(Seg a1),(Seg a2):],
bool [:(Seg a1),(Seg a2):] means :
Def2:
:: FINTOPO5:def 2
for b
1 being
set holds
( b
1 in [:(Seg a1),(Seg a2):] implies for b
2, b
3 being
Nat holds
( b
1 = [b2,b3] implies a
3 . b
1 = [:((Nbdl1 a1) . b2),((Nbdl1 a2) . b3):] ) );
existence
ex b1 being Function of [:(Seg c1),(Seg c2):], bool [:(Seg c1),(Seg c2):] st
for b2 being set holds
( b2 in [:(Seg c1),(Seg c2):] implies for b3, b4 being Nat holds
( b2 = [b3,b4] implies b1 . b2 = [:((Nbdl1 c1) . b3),((Nbdl1 c2) . b4):] ) )
uniqueness
for b1, b2 being Function of [:(Seg c1),(Seg c2):], bool [:(Seg c1),(Seg c2):] holds
( ( for b3 being set holds
( b3 in [:(Seg c1),(Seg c2):] implies for b4, b5 being Nat holds
( b3 = [b4,b5] implies b1 . b3 = [:((Nbdl1 c1) . b4),((Nbdl1 c2) . b5):] ) ) ) & ( for b3 being set holds
( b3 in [:(Seg c1),(Seg c2):] implies for b4, b5 being Nat holds
( b3 = [b4,b5] implies b2 . b3 = [:((Nbdl1 c1) . b4),((Nbdl1 c2) . b5):] ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Nbdl2 FINTOPO5:def 2 :
:: deftheorem Def3 defines FTSL2 FINTOPO5:def 3 :
theorem Th11: :: FINTOPO5:11
theorem Th12: :: FINTOPO5:12
theorem Th13: :: FINTOPO5:13
definition
let c
1, c
2 be
Nat;
func Nbds2 c
1,c
2 -> Function of
[:(Seg a1),(Seg a2):],
bool [:(Seg a1),(Seg a2):] means :
Def4:
:: FINTOPO5:def 4
for b
1 being
set holds
( b
1 in [:(Seg a1),(Seg a2):] implies for b
2, b
3 being
Nat holds
( b
1 = [b2,b3] implies a
3 . b
1 = [:{b2},((Nbdl1 a2) . b3):] \/ [:((Nbdl1 a1) . b2),{b3}:] ) );
existence
ex b1 being Function of [:(Seg c1),(Seg c2):], bool [:(Seg c1),(Seg c2):] st
for b2 being set holds
( b2 in [:(Seg c1),(Seg c2):] implies for b3, b4 being Nat holds
( b2 = [b3,b4] implies b1 . b2 = [:{b3},((Nbdl1 c2) . b4):] \/ [:((Nbdl1 c1) . b3),{b4}:] ) )
uniqueness
for b1, b2 being Function of [:(Seg c1),(Seg c2):], bool [:(Seg c1),(Seg c2):] holds
( ( for b3 being set holds
( b3 in [:(Seg c1),(Seg c2):] implies for b4, b5 being Nat holds
( b3 = [b4,b5] implies b1 . b3 = [:{b4},((Nbdl1 c2) . b5):] \/ [:((Nbdl1 c1) . b4),{b5}:] ) ) ) & ( for b3 being set holds
( b3 in [:(Seg c1),(Seg c2):] implies for b4, b5 being Nat holds
( b3 = [b4,b5] implies b2 . b3 = [:{b4},((Nbdl1 c2) . b5):] \/ [:((Nbdl1 c1) . b4),{b5}:] ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines Nbds2 FINTOPO5:def 4 :
for b
1, b
2 being
Natfor b
3 being
Function of
[:(Seg b1),(Seg b2):],
bool [:(Seg b1),(Seg b2):] holds
( b
3 = Nbds2 b
1,b
2 iff for b
4 being
set holds
( b
4 in [:(Seg b1),(Seg b2):] implies for b
5, b
6 being
Nat holds
( b
4 = [b5,b6] implies b
3 . b
4 = [:{b5},((Nbdl1 b2) . b6):] \/ [:((Nbdl1 b1) . b5),{b6}:] ) ) );
:: deftheorem Def5 defines FTSS2 FINTOPO5:def 5 :
theorem Th14: :: FINTOPO5:14
theorem Th15: :: FINTOPO5:15
theorem Th16: :: FINTOPO5:16