:: Completeness of the Lattices of Domains of a Topological Space
:: by Zbigniew Karno and Toshihiko Watanabe
::
:: Received July 16, 1992
:: Copyright (c) 1992-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies PRE_TOPC, SUBSET_1, TOPS_1, TARSKI, RCOMP_1, XBOOLE_0, SETFAM_1,
PCOMPS_1, ZFMISC_1, STRUCT_0, FINSET_1, FINSEQ_1, RELAT_1, NAT_1,
XXREAL_0, ARYTM_3, CARD_1, FUNCT_1, TDLAT_1, LATTICES, EQREL_1, PBOOLE,
LATTICE3, REWRITE1, TDLAT_2;
notations TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, SETFAM_1, STRUCT_0, FUNCT_1,
ORDINAL1, NUMBERS, FINSET_1, FINSEQ_1, XCMPLX_0, NAT_1, PRE_TOPC, TOPS_1,
TOPS_2, PCOMPS_1, BINOP_1, LATTICES, LATTICE3, TDLAT_1, XXREAL_0;
constructors SETFAM_1, XXREAL_0, NAT_1, MEMBERED, FINSEQ_1, TOPS_1, TOPS_2,
PCOMPS_1, LATTICE3, TDLAT_1, BINOP_1;
registrations XBOOLE_0, SUBSET_1, STRUCT_0, PRE_TOPC, TOPS_1, TDLAT_1,
XREAL_0, FINSEQ_1, RELAT_1, ORDINAL1, NAT_1;
requirements NUMERALS, BOOLE, SUBSET, ARITHM;
begin
:: 1. Preliminary Theorems about Subsets of a Toplogical Space.
reserve T for TopSpace;
theorem :: TDLAT_2:1
for A being Subset of T holds Int Cl Int A c= Int Cl A & Int Cl
Int A c= Cl Int A;
theorem :: TDLAT_2:2
for A being Subset of T holds Cl Int A c= Cl Int Cl A & Int Cl A
c= Cl Int Cl A;
theorem :: TDLAT_2:3
for A, B being Subset of T st B is closed holds Cl(Int(A /\ B)) = A
implies A c= B;
theorem :: TDLAT_2:4
for A, B being Subset of T st A is open holds Int(Cl(A \/ B)) = B
implies A c= B;
theorem :: TDLAT_2:5
for A being Subset of T st A c= Cl Int A holds A \/ Int Cl A c=
Cl Int(A \/ Int Cl A);
theorem :: TDLAT_2:6
for A being Subset of T st Int Cl A c= A holds Int Cl(A /\ Cl Int
A) c= A /\ Cl Int A;
begin
:: 2. The Closure and the Interior Operations for Families of Subsets of
:: a Topological Space.
::(for the definition of clf see PCOMPS_1:def 3)
notation
let T;
let F be Subset-Family of T;
synonym Cl F for clf F;
end;
::Properties of the Closure Operation Cl (see also PCOMPS_1).
theorem :: TDLAT_2:7
for F being Subset-Family of T holds Cl F = {A where A is Subset
of T : ex B being Subset of T st A = Cl B & B in F};
theorem :: TDLAT_2:8
for F being Subset-Family of T holds Cl F = Cl Cl F;
theorem :: TDLAT_2:9
for F being Subset-Family of T holds F = {} iff Cl F = {};
theorem :: TDLAT_2:10
for F,G being Subset-Family of T holds Cl(F /\ G) c= (Cl F) /\ (Cl G);
theorem :: TDLAT_2:11
for F,G being Subset-Family of T holds (Cl F) \ (Cl G) c= Cl(F \ G);
theorem :: TDLAT_2:12
for F being Subset-Family of T, A being Subset of T st A in F holds
meet(Cl F) c= Cl A & Cl A c= union(Cl F);
::for F being Subset-Family of T holds union F c= union(Cl F);
::(see PCOMPS_1:22)
theorem :: TDLAT_2:13
for F being Subset-Family of T holds meet F c= meet(Cl F);
theorem :: TDLAT_2:14
for F being Subset-Family of T holds Cl(meet F) c= meet(Cl F);
theorem :: TDLAT_2:15
for F being Subset-Family of T holds union(Cl F) c= Cl(union F);
definition
let T;
let F be Subset-Family of T;
func Int F -> Subset-Family of T means
:: TDLAT_2:def 1
for A being Subset of T holds
A in it iff ex B being Subset of T st A = Int B & B in F;
projectivity;
end;
::Properties of the Interior Operation Int.
theorem :: TDLAT_2:16
for F being Subset-Family of T holds Int F = {A where A is
Subset of T : ex B being Subset of T st A = Int B & B in F};
::$CT
theorem :: TDLAT_2:18
for F being Subset-Family of T holds Int F is open;
theorem :: TDLAT_2:19
for F being Subset-Family of T holds F = {} iff Int F = {};
theorem :: TDLAT_2:20
for A being Subset of T, F being Subset-Family of T st F = { A }
holds Int F = { Int A };
theorem :: TDLAT_2:21
for F,G being Subset-Family of T holds F c= G implies Int F c= Int G;
theorem :: TDLAT_2:22
for F,G being Subset-Family of T holds Int(F \/ G) = (Int F) \/ (Int G);
theorem :: TDLAT_2:23
for F,G being Subset-Family of T holds Int(F /\ G) c= (Int F) /\ (Int G);
theorem :: TDLAT_2:24
for F,G being Subset-Family of T holds (Int F) \ (Int G) c= Int(F \ G);
theorem :: TDLAT_2:25
for F being Subset-Family of T, A being Subset of T holds A in F
implies Int A c= union(Int F) & meet(Int F) c= Int A;
theorem :: TDLAT_2:26
for F being Subset-Family of T holds union(Int F) c= union F;
theorem :: TDLAT_2:27
for F being Subset-Family of T holds meet(Int F) c= meet F;
theorem :: TDLAT_2:28
for F being Subset-Family of T holds union(Int F) c= Int(union F );
theorem :: TDLAT_2:29
for F being Subset-Family of T holds Int(meet F) c= meet(Int F);
theorem :: TDLAT_2:30
for F being Subset-Family of T holds F is finite implies Int(meet F) =
meet(Int F);
::Connections between the Operations Int and Cl.
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
theorem :: TDLAT_2:31
Cl Int F = {A where A is Subset of T : ex B being Subset of T st
A = Cl Int B & B in F};
theorem :: TDLAT_2:32
Int Cl F = {A where A is Subset of T : ex B being Subset of T st
A = Int Cl B & B in F};
theorem :: TDLAT_2:33
Cl Int Cl F = {A where A is Subset of T : ex B being Subset of T
st A = Cl Int Cl B & B in F};
theorem :: TDLAT_2:34
Int Cl Int F = {A where A is Subset of T : ex B being Subset of
T st A = Int Cl Int B & B in F};
theorem :: TDLAT_2:35
Cl Int Cl Int F = Cl Int F;
theorem :: TDLAT_2:36
Int Cl Int Cl F = Int Cl F;
theorem :: TDLAT_2:37
union(Int Cl F) c= union(Cl Int Cl F);
theorem :: TDLAT_2:38
meet(Int Cl F) c= meet(Cl Int Cl F);
theorem :: TDLAT_2:39
union(Cl Int F) c= union(Cl Int Cl F);
theorem :: TDLAT_2:40
meet(Cl Int F) c= meet(Cl Int Cl F);
theorem :: TDLAT_2:41
union(Int Cl Int F) c= union(Int Cl F);
theorem :: TDLAT_2:42
meet(Int Cl Int F) c= meet(Int Cl F);
theorem :: TDLAT_2:43
union(Int Cl Int F) c= union(Cl Int F);
theorem :: TDLAT_2:44
meet(Int Cl Int F) c= meet(Cl Int F);
theorem :: TDLAT_2:45
union(Cl Int Cl F) c= union(Cl F);
theorem :: TDLAT_2:46
meet(Cl Int Cl F) c= meet(Cl F);
theorem :: TDLAT_2:47
union(Int F) c= union(Int Cl Int F);
theorem :: TDLAT_2:48
meet(Int F) c= meet(Int Cl Int F);
theorem :: TDLAT_2:49
union(Cl Int F) c= Cl Int(union F);
theorem :: TDLAT_2:50
Cl Int(meet F) c= meet(Cl Int F);
theorem :: TDLAT_2:51
union(Int Cl F) c= Int Cl(union F);
theorem :: TDLAT_2:52
Int Cl(meet F) c= meet(Int Cl F);
theorem :: TDLAT_2:53
union(Cl Int Cl F) c= Cl Int Cl(union F);
theorem :: TDLAT_2:54
Cl Int Cl(meet F) c= meet(Cl Int Cl F);
theorem :: TDLAT_2:55
union(Int Cl Int F) c= Int Cl Int(union F);
theorem :: TDLAT_2:56
Int Cl Int(meet F) c= meet(Int Cl Int F);
theorem :: TDLAT_2:57
for F being Subset-Family of T holds (for A being Subset of T st
A in F holds A c= Cl Int A) implies union F c= Cl Int(union F) & Cl(union F) =
Cl Int Cl(union F);
theorem :: TDLAT_2:58
for F being Subset-Family of T holds (for A being Subset of T st
A in F holds Int Cl A c= A) implies Int Cl(meet F) c= meet F & Int Cl Int(meet
F) = Int(meet F);
begin
:: 3. Selected Properties of Domains of a Topological Space.
reserve T for non empty TopSpace;
theorem :: TDLAT_2:59
for A, B being Subset of T st B is condensed holds Int(Cl(A \/ B
)) \/ (A \/ B) = B iff A c= B;
theorem :: TDLAT_2:60
for A, B being Subset of T st A is condensed holds Cl(Int(A /\ B)) /\
(A /\ B) = A iff A c= B;
theorem :: TDLAT_2:61
for A, B being Subset of T st A is closed_condensed & B is
closed_condensed holds Int A c= Int B iff A c= B;
theorem :: TDLAT_2:62
for A, B being Subset of T st A is open_condensed & B is
open_condensed holds Cl A c= Cl B iff A c= B;
theorem :: TDLAT_2:63
for A, B being Subset of T st A is closed_condensed holds A c= B
implies Cl(Int(A /\ B)) = A;
theorem :: TDLAT_2:64
for A, B being Subset of T st B is open_condensed holds A c= B
implies Int(Cl(A \/ B)) = B;
definition
let T;
let IT be Subset-Family of T;
attr IT is domains-family means
:: TDLAT_2:def 2
for A being Subset of T holds A in IT implies A is condensed;
end;
theorem :: TDLAT_2:65
for F being Subset-Family of T holds F c= Domains_of T iff F is
domains-family;
theorem :: TDLAT_2:66
for F being Subset-Family of T holds F is domains-family implies
union F c= Cl Int(union F) & Cl(union F) = Cl Int Cl(union F);
theorem :: TDLAT_2:67
for F being Subset-Family of T holds F is domains-family implies
Int Cl(meet F) c= meet F & Int Cl Int(meet F) = Int(meet F);
theorem :: TDLAT_2:68
for F being Subset-Family of T holds F is domains-family implies
(union F) \/ (Int Cl(union F)) is condensed;
theorem :: TDLAT_2:69
for F being Subset-Family of T holds (for B being Subset of T st
B in F holds B c= (union F) \/ (Int Cl(union F))) & for A being Subset of T st
A is condensed holds (for B being Subset of T st B in F holds B c= A) implies (
union F) \/ (Int Cl(union F)) c= A;
theorem :: TDLAT_2:70
for F being Subset-Family of T holds F is domains-family implies
(meet F) /\ (Cl Int(meet F)) is condensed;
theorem :: TDLAT_2:71
for F being Subset-Family of T holds (for B being Subset of T st
B in F holds (meet F) /\ (Cl Int(meet F)) c= B) & (F = {} or for A being Subset
of T st A is condensed holds (for B being Subset of T st B in F holds A c= B)
implies A c= (meet F) /\ (Cl Int(meet F)));
definition
let T;
let IT be Subset-Family of T;
attr IT is closed-domains-family means
:: TDLAT_2:def 3
for A being Subset of T holds A in IT implies A is closed_condensed;
end;
theorem :: TDLAT_2:72
for F being Subset-Family of T holds F c= Closed_Domains_of T
iff F is closed-domains-family;
theorem :: TDLAT_2:73
for F being Subset-Family of T holds F is closed-domains-family
implies F is domains-family;
theorem :: TDLAT_2:74
for F being Subset-Family of T holds F is closed-domains-family
implies F is closed;
theorem :: TDLAT_2:75
for F being Subset-Family of T holds F is domains-family implies Cl F
is closed-domains-family;
theorem :: TDLAT_2:76
for F being Subset-Family of T holds F is closed-domains-family
implies Cl(union F) is closed_condensed & Cl Int(meet F) is closed_condensed;
theorem :: TDLAT_2:77
for F being Subset-Family of T holds (for B being Subset of T st
B in F holds B c= Cl(union F)) & for A being Subset of T st A is
closed_condensed holds (for B being Subset of T st B in F holds B c= A) implies
Cl(union F) c= A;
theorem :: TDLAT_2:78
for F being Subset-Family of T holds (F is closed implies for B
being Subset of T st B in F holds Cl Int(meet F) c= B) & (F = {} or for A being
Subset of T st A is closed_condensed holds (for B being Subset of T st B in F
holds A c= B) implies A c= Cl Int(meet F));
definition
let T;
let IT be Subset-Family of T;
attr IT is open-domains-family means
:: TDLAT_2:def 4
for A being Subset of T holds A in IT implies A is open_condensed;
end;
theorem :: TDLAT_2:79
for F being Subset-Family of T holds F c= Open_Domains_of T iff
F is open-domains-family;
theorem :: TDLAT_2:80
for F being Subset-Family of T holds F is open-domains-family
implies F is domains-family;
theorem :: TDLAT_2:81
for F being Subset-Family of T holds F is open-domains-family
implies F is open;
theorem :: TDLAT_2:82
for F being Subset-Family of T holds F is domains-family implies Int F
is open-domains-family;
theorem :: TDLAT_2:83
for F being Subset-Family of T holds F is open-domains-family
implies Int(meet F) is open_condensed & Int Cl(union F) is open_condensed;
theorem :: TDLAT_2:84
for F being Subset-Family of T holds (F is open implies for B
being Subset of T st B in F holds B c= Int Cl(union F)) & for A being Subset of
T st A is open_condensed holds (for B being Subset of T st B in F holds B c= A)
implies Int Cl(union F) c= A;
theorem :: TDLAT_2:85
for F being Subset-Family of T holds (for B being Subset of T st
B in F holds Int(meet F) c= B) & (F = {} or for A being Subset of T st A is
open_condensed holds (for B being Subset of T st B in F holds A c= B) implies A
c= Int(meet F));
begin
:: 4. Completeness of the Lattice of Domains.
reserve T for non empty TopSpace;
theorem :: TDLAT_2:86
the carrier of Domains_Lattice T = Domains_of T;
theorem :: TDLAT_2:87
for a, b being Element of Domains_Lattice T for A, B being
Element of Domains_of T st a = A & b = B holds a "\/" b = Int(Cl(A \/ B)) \/ (A
\/ B) & a "/\" b = Cl(Int(A /\ B)) /\ (A /\ B);
theorem :: TDLAT_2:88
Bottom (Domains_Lattice T) = {}T & Top (Domains_Lattice T) = [#]T;
theorem :: TDLAT_2:89
for a, b being Element of Domains_Lattice T for A, B being
Element of Domains_of T st a = A & b = B holds a [= b iff A c= B;
theorem :: TDLAT_2:90
for X being Subset of Domains_Lattice T ex a being Element of
Domains_Lattice T st X is_less_than a & for b being Element of Domains_Lattice
T st X is_less_than b holds a [= b;
theorem :: TDLAT_2:91
Domains_Lattice T is complete;
theorem :: TDLAT_2:92
for F being Subset-Family of T st F is domains-family for X
being Subset of Domains_Lattice T st X = F holds "\/"(X,Domains_Lattice T) = (
union F) \/ (Int Cl(union F));
theorem :: TDLAT_2:93
for F being Subset-Family of T st F is domains-family for X
being Subset of Domains_Lattice T st X = F holds (X <> {} implies "/\"(X,
Domains_Lattice T) = (meet F) /\ (Cl Int(meet F))) & (X = {} implies "/\"(X,
Domains_Lattice T) = [#]T);
begin
:: 5. Completeness of the Lattices of Closed Domains and Open Domains.
reserve T for non empty TopSpace;
::The Lattice of Closed Domains.
theorem :: TDLAT_2:94
the carrier of Closed_Domains_Lattice T = Closed_Domains_of T;
theorem :: TDLAT_2:95
for a, b being Element of Closed_Domains_Lattice T for A, B
being Element of Closed_Domains_of T st a = A & b = B holds a "\/" b = A \/ B &
a "/\" b = Cl(Int(A /\ B));
theorem :: TDLAT_2:96
Bottom (Closed_Domains_Lattice T) = {}T & Top (Closed_Domains_Lattice
T) = [#]T;
theorem :: TDLAT_2:97
for a, b being Element of Closed_Domains_Lattice T for A, B
being Element of Closed_Domains_of T st a = A & b = B holds a [= b iff A c= B
;
theorem :: TDLAT_2:98
for X being Subset of Closed_Domains_Lattice T ex a being
Element of Closed_Domains_Lattice T st X is_less_than a & for b being Element
of Closed_Domains_Lattice T st X is_less_than b holds a [= b;
theorem :: TDLAT_2:99
Closed_Domains_Lattice T is complete;
theorem :: TDLAT_2:100
for F being Subset-Family of T st F is closed-domains-family for X
being Subset of Closed_Domains_Lattice T st X = F holds "\/"(X,
Closed_Domains_Lattice T) = Cl(union F);
theorem :: TDLAT_2:101
for F being Subset-Family of T st F is closed-domains-family for X
being Subset of Closed_Domains_Lattice T st X = F holds (X <> {} implies "/\"(X
,Closed_Domains_Lattice T) = Cl(Int(meet F))) & (X = {} implies "/\"(X,
Closed_Domains_Lattice T) = [#]T);
theorem :: TDLAT_2:102
for F being Subset-Family of T st F is closed-domains-family for X
being Subset of Domains_Lattice T st X = F holds (X <> {} implies "/\"(X,
Domains_Lattice T) = Cl(Int(meet F))) & (X = {} implies "/\"(X,Domains_Lattice
T) = [#]T);
::The Lattice of Open Domains.
theorem :: TDLAT_2:103
the carrier of Open_Domains_Lattice T = Open_Domains_of T;
theorem :: TDLAT_2:104
for a, b being Element of Open_Domains_Lattice T for A, B being
Element of Open_Domains_of T st a = A & b = B holds a "\/" b = Int(Cl(A \/ B))
& a "/\" b = A /\ B;
theorem :: TDLAT_2:105
Bottom (Open_Domains_Lattice T) = {}T & Top (Open_Domains_Lattice T) = [#] T;
theorem :: TDLAT_2:106
for a, b being Element of Open_Domains_Lattice T for A, B being
Element of Open_Domains_of T st a = A & b = B holds a [= b iff A c= B;
theorem :: TDLAT_2:107
for X being Subset of Open_Domains_Lattice T ex a being Element
of Open_Domains_Lattice T st X is_less_than a & for b being Element of
Open_Domains_Lattice T st X is_less_than b holds a [= b;
theorem :: TDLAT_2:108
Open_Domains_Lattice T is complete;
theorem :: TDLAT_2:109
for F being Subset-Family of T st F is open-domains-family for X being
Subset of Open_Domains_Lattice T st X = F holds "\/"(X,Open_Domains_Lattice T)
= Int Cl(union F);
theorem :: TDLAT_2:110
for F being Subset-Family of T st F is open-domains-family for X being
Subset of Open_Domains_Lattice T st X = F holds (X <> {} implies "/\"(X,
Open_Domains_Lattice T) = Int(meet F)) & (X = {} implies "/\"(X,
Open_Domains_Lattice T) = [#]T);
theorem :: TDLAT_2:111
for F being Subset-Family of T st F is open-domains-family for X being
Subset of Domains_Lattice T st X = F holds "\/"(X,Domains_Lattice T) = Int Cl(
union F);