Article PRE_TOPC, MML version 4.99.1005
:: PRE_TOPC:structnot 1 => PRE_TOPC:struct 1
definition
struct(1-sorted) TopStruct(#
carrier -> set,
topology -> Element of bool bool the carrier of it
#);
end;
:: PRE_TOPC:attrnot 1 => PRE_TOPC:attr 1
definition
let a1 be TopStruct;
attr a1 is strict;
end;
:: PRE_TOPC:exreg 1
registration
cluster strict TopStruct;
end;
:: PRE_TOPC:aggrnot 1 => PRE_TOPC:aggr 1
definition
let a1 be set;
let a2 be Element of bool bool a1;
aggr TopStruct(#a1,a2#) -> strict TopStruct;
end;
:: PRE_TOPC:selnot 1 => PRE_TOPC:sel 1
definition
let a1 be TopStruct;
sel the topology of a1 -> Element of bool bool the carrier of a1;
end;
:: PRE_TOPC:attrnot 2 => PRE_TOPC:attr 2
definition
let a1 be TopStruct;
attr a1 is TopSpace-like means
the carrier of a1 in the topology of a1 &
(for b1 being Element of bool bool the carrier of a1
st b1 c= the topology of a1
holds union b1 in the topology of a1) &
(for b1, b2 being Element of bool the carrier of a1
st b1 in the topology of a1 & b2 in the topology of a1
holds b1 /\ b2 in the topology of a1);
end;
:: PRE_TOPC:dfs 1
definiens
let a1 be TopStruct;
To prove
a1 is TopSpace-like
it is sufficient to prove
thus the carrier of a1 in the topology of a1 &
(for b1 being Element of bool bool the carrier of a1
st b1 c= the topology of a1
holds union b1 in the topology of a1) &
(for b1, b2 being Element of bool the carrier of a1
st b1 in the topology of a1 & b2 in the topology of a1
holds b1 /\ b2 in the topology of a1);
:: PRE_TOPC:def 1
theorem
for b1 being TopStruct holds
b1 is TopSpace-like
iff
the carrier of b1 in the topology of b1 &
(for b2 being Element of bool bool the carrier of b1
st b2 c= the topology of b1
holds union b2 in the topology of b1) &
(for b2, b3 being Element of bool the carrier of b1
st b2 in the topology of b1 & b3 in the topology of b1
holds b2 /\ b3 in the topology of b1);
:: PRE_TOPC:exreg 2
registration
cluster non empty strict TopSpace-like TopStruct;
end;
:: PRE_TOPC:modenot 1
definition
mode TopSpace is TopSpace-like TopStruct;
end;
:: PRE_TOPC:modenot 2
definition
let a1 be 1-sorted;
mode Point of a1 is Element of the carrier of a1;
end;
:: PRE_TOPC:funcreg 1
registration
let a1 be TopSpace-like TopStruct;
cluster the topology of a1 -> non empty;
end;
:: PRE_TOPC:th 5
theorem
for b1 being TopSpace-like TopStruct holds
{} in the topology of b1;
:: PRE_TOPC:th 12
theorem
for b1 being 1-sorted holds
[#] b1 = the carrier of b1;
:: PRE_TOPC:th 13
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1 holds
b2 in [#] b1;
:: PRE_TOPC:th 14
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
b2 c= [#] b1;
:: PRE_TOPC:th 16
theorem
for b1 being 1-sorted
for b2 being set
st b2 c= [#] b1
holds b2 is Element of bool the carrier of b1;
:: PRE_TOPC:th 17
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
b2 ` = ([#] b1) \ b2;
:: PRE_TOPC:th 18
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
b2 \/ (b2 `) = [#] b1;
:: PRE_TOPC:th 22
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
([#] b1) \ (([#] b1) \ b2) = b2;
:: PRE_TOPC:th 23
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
b2 <> [#] b1
iff
([#] b1) \ b2 <> {};
:: PRE_TOPC:th 25
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool the carrier of b1
st [#] b1 = b2 \/ b3 & b2 misses b3
holds b3 = ([#] b1) \ b2;
:: PRE_TOPC:th 27
theorem
for b1 being 1-sorted holds
[#] b1 = ({} b1) `;
:: PRE_TOPC:attrnot 3 => PRE_TOPC:attr 3
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is open means
a2 in the topology of a1;
end;
:: PRE_TOPC:dfs 2
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is open
it is sufficient to prove
thus a2 in the topology of a1;
:: PRE_TOPC:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
b2 in the topology of b1;
:: PRE_TOPC:attrnot 4 => PRE_TOPC:attr 4
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is closed means
([#] a1) \ a2 is open(a1);
end;
:: PRE_TOPC:dfs 3
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is closed
it is sufficient to prove
thus ([#] a1) \ a2 is open(a1);
:: PRE_TOPC:def 6
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
([#] b1) \ b2 is open(b1);
:: PRE_TOPC:prednot 1 => PRE_TOPC:pred 1
definition
let a1 be 1-sorted;
let a2 be Element of bool bool the carrier of a1;
pred A2 is_a_cover_of A1 means
[#] a1 = union a2;
end;
:: PRE_TOPC:dfs 4
definiens
let a1 be 1-sorted;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is_a_cover_of a1
it is sufficient to prove
thus [#] a1 = union a2;
:: PRE_TOPC:def 8
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
b2 is_a_cover_of b1
iff
[#] b1 = union b2;
:: PRE_TOPC:modenot 3 => PRE_TOPC:mode 1
definition
let a1 be TopStruct;
mode SubSpace of A1 -> TopStruct means
[#] it c= [#] a1 &
(for b1 being Element of bool the carrier of it holds
b1 in the topology of it
iff
ex b2 being Element of bool the carrier of a1 st
b2 in the topology of a1 & b1 = b2 /\ [#] it);
end;
:: PRE_TOPC:dfs 5
definiens
let a1, a2 be TopStruct;
To prove
a2 is SubSpace of a1
it is sufficient to prove
thus [#] a2 c= [#] a1 &
(for b1 being Element of bool the carrier of a2 holds
b1 in the topology of a2
iff
ex b2 being Element of bool the carrier of a1 st
b2 in the topology of a1 & b1 = b2 /\ [#] a2);
:: PRE_TOPC:def 9
theorem
for b1, b2 being TopStruct holds
b2 is SubSpace of b1
iff
[#] b2 c= [#] b1 &
(for b3 being Element of bool the carrier of b2 holds
b3 in the topology of b2
iff
ex b4 being Element of bool the carrier of b1 st
b4 in the topology of b1 & b3 = b4 /\ [#] b2);
:: PRE_TOPC:exreg 3
registration
let a1 be TopStruct;
cluster strict SubSpace of a1;
end;
:: PRE_TOPC:exreg 4
registration
let a1 be non empty TopStruct;
cluster non empty strict SubSpace of a1;
end;
:: PRE_TOPC:condreg 1
registration
let a1 be TopSpace-like TopStruct;
cluster -> TopSpace-like (SubSpace of a1);
end;
:: PRE_TOPC:funcnot 1 => PRE_TOPC:func 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
func A1 | A2 -> strict SubSpace of a1 means
[#] it = a2;
end;
:: PRE_TOPC:def 10
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being strict SubSpace of b1 holds
b3 = b1 | b2
iff
[#] b3 = b2;
:: PRE_TOPC:funcreg 2
registration
let a1 be non empty TopStruct;
let a2 be non empty Element of bool the carrier of a1;
cluster a1 | a2 -> non empty strict;
end;
:: PRE_TOPC:exreg 5
registration
let a1 be TopSpace-like TopStruct;
cluster strict TopSpace-like SubSpace of a1;
end;
:: PRE_TOPC:funcreg 3
registration
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
cluster a1 | a2 -> strict TopSpace-like;
end;
:: PRE_TOPC:th 28
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b1 | b3
st b2 = b4 & b2 c= b3
holds b1 | b2 = (b1 | b3) | b4;
:: PRE_TOPC:th 29
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
the carrier of b1 | b2 = b2;
:: PRE_TOPC:th 30
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1 holds
b3 | b4 is Function-like quasi_total Relation of the carrier of b1 | b4,the carrier of b2;
:: PRE_TOPC:attrnot 5 => PRE_TOPC:attr 5
definition
let a1, a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is continuous means
for b1 being Element of bool the carrier of a2
st b1 is closed(a2)
holds a3 " b1 is closed(a1);
end;
:: PRE_TOPC:dfs 7
definiens
let a1, a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is continuous
it is sufficient to prove
thus for b1 being Element of bool the carrier of a2
st b1 is closed(a2)
holds a3 " b1 is closed(a1);
:: PRE_TOPC:def 12
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is continuous(b1, b2)
iff
for b4 being Element of bool the carrier of b2
st b4 is closed(b2)
holds b3 " b4 is closed(b1);
:: PRE_TOPC:th 31
theorem
for b1, b2, b3, b4 being TopStruct
st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of b4,the topology of b4#) &
b3 is SubSpace of b1
holds b4 is SubSpace of b2;
:: PRE_TOPC:th 39
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b2 holds
b3 is Element of bool the carrier of b1;
:: PRE_TOPC:th 41
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 <> {} b1
holds ex b3 being Element of the carrier of b1 st
b3 in b2;
:: PRE_TOPC:th 42
theorem
for b1 being TopSpace-like TopStruct holds
[#] b1 is closed(b1);
:: PRE_TOPC:funcreg 4
registration
let a1 be TopSpace-like TopStruct;
cluster [#] a1 -> closed;
end;
:: PRE_TOPC:exreg 6
registration
let a1 be TopSpace-like TopStruct;
cluster closed Element of bool the carrier of a1;
end;
:: PRE_TOPC:exreg 7
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty closed Element of bool the carrier of a1;
end;
:: PRE_TOPC:th 43
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b2 holds
b3 is closed(b2)
iff
ex b4 being Element of bool the carrier of b1 st
b4 is closed(b1) & b4 /\ [#] b2 = b3;
:: PRE_TOPC:th 44
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is closed(b1)
holds meet b2 is closed(b1);
:: PRE_TOPC:funcnot 2 => PRE_TOPC:func 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
func Cl A2 -> Element of bool the carrier of a1 means
for b1 being set
st b1 in the carrier of a1
holds b1 in it
iff
for b2 being Element of bool the carrier of a1
st b2 is open(a1) & b1 in b2
holds a2 meets b2;
projectivity;
:: for a1 being TopStruct
:: for a2 being Element of bool the carrier of a1 holds
:: Cl Cl a2 = Cl a2;
end;
:: PRE_TOPC:def 13
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b3 = Cl b2
iff
for b4 being set
st b4 in the carrier of b1
holds b4 in b3
iff
for b5 being Element of bool the carrier of b1
st b5 is open(b1) & b4 in b5
holds b2 meets b5;
:: PRE_TOPC:th 45
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set
st b3 in the carrier of b1
holds b3 in Cl b2
iff
for b4 being Element of bool the carrier of b1
st b4 is closed(b1) & b2 c= b4
holds b3 in b4;
:: PRE_TOPC:th 46
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
ex b3 being Element of bool bool the carrier of b1 st
(for b4 being Element of bool the carrier of b1 holds
b4 in b3
iff
b4 is closed(b1) & b2 c= b4) &
Cl b2 = meet b3;
:: PRE_TOPC:th 47
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds Cl b4 = (Cl b3) /\ [#] b2;
:: PRE_TOPC:th 48
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 c= Cl b2;
:: PRE_TOPC:th 49
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds Cl b2 c= Cl b3;
:: PRE_TOPC:th 50
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Cl (b2 \/ b3) = (Cl b2) \/ Cl b3;
:: PRE_TOPC:th 51
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Cl (b2 /\ b3) c= (Cl b2) /\ Cl b3;
:: PRE_TOPC:th 52
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
(b2 is closed(b1) implies Cl b2 = b2) &
(b1 is TopSpace-like & Cl b2 = b2 implies b2 is closed(b1));
:: PRE_TOPC:th 53
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
(b2 is open(b1) implies Cl (([#] b1) \ b2) = ([#] b1) \ b2) &
(b1 is TopSpace-like &
Cl (([#] b1) \ b2) = ([#] b1) \ b2 implies b2 is open(b1));
:: PRE_TOPC:th 54
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in Cl b2
iff
b1 is not empty &
(for b4 being Element of bool the carrier of b1
st b4 is open(b1) & b3 in b4
holds b2 meets b4);