:: FINTOPO5 semantic presentation
theorem Th1: :: FINTOPO5:1
theorem Th2: :: FINTOPO5:2
:: 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 n be
Element of
NAT ,
m be
Element of
NAT ;
func Nbdl2 c1,
c2 -> Relation of
[:(Seg a1),(Seg a2):] means :
Def2:
:: FINTOPO5:def 2
for
x being
set st
x in [:(Seg n),(Seg m):] holds
for
i,
j being
Element of
NAT st
x = [i,j] holds
it .: {x} = [:((Nbdl1 n) .: {i}),((Nbdl1 m) .: {j}):];
existence
ex b1 being Relation of [:(Seg n),(Seg m):] st
for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
b1 .: {x} = [:((Nbdl1 n) .: {i}),((Nbdl1 m) .: {j}):]
uniqueness
for b1, b2 being Relation of [:(Seg n),(Seg m):] st ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
b1 .: {x} = [:((Nbdl1 n) .: {i}),((Nbdl1 m) .: {j}):] ) & ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
b2 .: {x} = [:((Nbdl1 n) .: {i}),((Nbdl1 m) .: {j}):] ) holds
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 n be
Element of
NAT ,
m be
Element of
NAT ;
func Nbds2 c1,
c2 -> Relation of
[:(Seg a1),(Seg a2):] means :
Def4:
:: FINTOPO5:def 4
for
x being
set st
x in [:(Seg n),(Seg m):] holds
for
i,
j being
Element of
NAT st
x = [i,j] holds
it .: {x} = [:{i},((Nbdl1 m) .: {j}):] \/ [:((Nbdl1 n) .: {i}),{j}:];
existence
ex b1 being Relation of [:(Seg n),(Seg m):] st
for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
b1 .: {x} = [:{i},((Nbdl1 m) .: {j}):] \/ [:((Nbdl1 n) .: {i}),{j}:]
uniqueness
for b1, b2 being Relation of [:(Seg n),(Seg m):] st ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
b1 .: {x} = [:{i},((Nbdl1 m) .: {j}):] \/ [:((Nbdl1 n) .: {i}),{j}:] ) & ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
b2 .: {x} = [:{i},((Nbdl1 m) .: {j}):] \/ [:((Nbdl1 n) .: {i}),{j}:] ) holds
b1 = b2
end;
:: deftheorem Def4 defines Nbds2 FINTOPO5:def 4 :
:: deftheorem Def5 defines FTSS2 FINTOPO5:def 5 :
theorem Th14: :: FINTOPO5:14
theorem Th15: :: FINTOPO5:15
theorem Th16: :: FINTOPO5:16