Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### The Lattice of Domains of an Extremally Disconnected Space

**by****Zbigniew Karno**- Received August 27, 1992
- MML identifier: TDLAT_3

- [ Mizar article, MML identifier index ]

environ vocabulary PRE_TOPC, TOPS_1, BOOLE, SUBSET_1, NATTRA_1, SETFAM_1, TARSKI, TDLAT_1, RELAT_1, LATTICES, FUNCT_1, TDLAT_2, TDLAT_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, STRUCT_0, RELAT_1, PRE_TOPC, TOPS_1, TOPS_2, PCOMPS_1, BORSUK_1, TSEP_1, BINOP_1, LATTICES, LATTICE3, TDLAT_1, TDLAT_2; constructors TOPS_1, TOPS_2, BORSUK_1, TSEP_1, LATTICE3, TDLAT_1, TDLAT_2, PARTFUN1, MEMBERED; clusters PRE_TOPC, BORSUK_1, TSEP_1, LATTICES, STRUCT_0, COMPLSP1, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1; requirements BOOLE, SUBSET; begin :: 1. Selected Properties of Subsets of a Topological Space. reserve X for TopSpace; ::Properties of the Closure and the Interior Operators. reserve C for Subset of X; canceled; theorem :: TDLAT_3:2 Cl C = (Int C`)`; theorem :: TDLAT_3:3 Cl C` = (Int C)`; theorem :: TDLAT_3:4 Int C` = (Cl C)`; reserve A, B for Subset of X; canceled; theorem :: TDLAT_3:6 A \/ B = the carrier of X implies (A is closed implies A \/ Int B = the carrier of X) & (B is closed implies Int A \/ B = the carrier of X); theorem :: TDLAT_3:7 A is open & A is closed iff Cl A = Int A; theorem :: TDLAT_3:8 A is open & A is closed implies Int Cl A = Cl Int A; ::Properties of Domains. theorem :: TDLAT_3:9 A is condensed & Cl Int A c= Int Cl A implies A is open_condensed & A is closed_condensed; theorem :: TDLAT_3:10 A is condensed & Cl Int A c= Int Cl A implies A is open & A is closed; theorem :: TDLAT_3:11 A is condensed implies Int Cl A = Int A & Cl A = Cl Int A; begin :: 2. Discrete Topological Structures. definition let IT be TopStruct; attr IT is discrete means :: TDLAT_3:def 1 the topology of IT = bool the carrier of IT; attr IT is anti-discrete means :: TDLAT_3:def 2 the topology of IT = {{}, the carrier of IT}; end; theorem :: TDLAT_3:12 for Y being TopStruct holds Y is discrete & Y is anti-discrete implies bool the carrier of Y = {{}, the carrier of Y}; theorem :: TDLAT_3:13 for Y being TopStruct st {} in the topology of Y & the carrier of Y in the topology of Y holds bool the carrier of Y = {{}, the carrier of Y} implies Y is discrete & Y is anti-discrete; definition cluster discrete anti-discrete strict non empty TopStruct; end; theorem :: TDLAT_3:14 for Y being discrete TopStruct, A being Subset of Y holds (the carrier of Y) \ A in the topology of Y; theorem :: TDLAT_3:15 for Y being anti-discrete TopStruct, A being Subset of Y st A in the topology of Y holds (the carrier of Y) \ A in the topology of Y; definition cluster discrete -> TopSpace-like TopStruct; cluster anti-discrete -> TopSpace-like TopStruct; end; theorem :: TDLAT_3:16 for Y being TopSpace-like TopStruct holds bool the carrier of Y = {{}, the carrier of Y} implies Y is discrete & Y is anti-discrete; definition let IT be TopStruct; attr IT is almost_discrete means :: TDLAT_3:def 3 for A being Subset of IT st A in the topology of IT holds (the carrier of IT) \ A in the topology of IT; end; definition cluster discrete -> almost_discrete TopStruct; cluster anti-discrete -> almost_discrete TopStruct; end; definition cluster almost_discrete strict TopStruct; end; begin :: 3. Discrete Topological Spaces. definition cluster discrete anti-discrete strict non empty TopSpace; end; theorem :: TDLAT_3:17 X is discrete iff for A being Subset of X holds A is open; theorem :: TDLAT_3:18 X is discrete iff for A being Subset of X holds A is closed; theorem :: TDLAT_3:19 (for A being Subset of X, x being Point of X st A = {x} holds A is open) implies X is discrete; definition let X be discrete non empty TopSpace; cluster -> open closed discrete SubSpace of X; end; definition let X be discrete non empty TopSpace; cluster discrete strict SubSpace of X; end; theorem :: TDLAT_3:20 X is anti-discrete iff for A being Subset of X st A is open holds A = {} or A = the carrier of X; theorem :: TDLAT_3:21 X is anti-discrete iff for A being Subset of X st A is closed holds A = {} or A = the carrier of X; theorem :: TDLAT_3:22 (for A being Subset of X, x being Point of X st A = {x} holds Cl A = the carrier of X) implies X is anti-discrete; definition let X be anti-discrete non empty TopSpace; cluster -> anti-discrete SubSpace of X; end; definition let X be anti-discrete non empty TopSpace; cluster anti-discrete SubSpace of X; end; theorem :: TDLAT_3:23 X is almost_discrete iff for A being Subset of X st A is open holds A is closed; theorem :: TDLAT_3:24 X is almost_discrete iff for A being Subset of X st A is closed holds A is open; theorem :: TDLAT_3:25 X is almost_discrete iff for A being Subset of X st A is open holds Cl A = A; theorem :: TDLAT_3:26 X is almost_discrete iff for A being Subset of X st A is closed holds Int A = A; definition cluster almost_discrete strict TopSpace; end; theorem :: TDLAT_3:27 (for A being Subset of X, x being Point of X st A = {x} holds Cl A is open) implies X is almost_discrete; theorem :: TDLAT_3:28 X is discrete iff X is almost_discrete & (for A being Subset of X, x being Point of X st A = {x} holds A is closed); definition cluster discrete -> almost_discrete TopSpace; cluster anti-discrete -> almost_discrete TopSpace; end; definition let X be almost_discrete non empty TopSpace; cluster -> almost_discrete (non empty SubSpace of X); end; definition let X be almost_discrete non empty TopSpace; cluster open -> closed SubSpace of X; cluster closed -> open SubSpace of X; end; definition let X be almost_discrete non empty TopSpace; cluster almost_discrete strict non empty SubSpace of X; end; begin :: 4. Extremally Disconnected Topological Spaces. definition let IT be non empty TopSpace; attr IT is extremally_disconnected means :: TDLAT_3:def 4 for A being Subset of IT st A is open holds Cl A is open; end; definition cluster extremally_disconnected strict (non empty TopSpace); end; reserve X for non empty TopSpace; theorem :: TDLAT_3:29 X is extremally_disconnected iff for A being Subset of X st A is closed holds Int A is closed; theorem :: TDLAT_3:30 X is extremally_disconnected iff for A, B being Subset of X st A is open & B is open holds A misses B implies Cl A misses Cl B; theorem :: TDLAT_3:31 X is extremally_disconnected iff for A, B being Subset of X st A is closed & B is closed holds A \/ B = the carrier of X implies (Int A) \/ (Int B) = the carrier of X; theorem :: TDLAT_3:32 X is extremally_disconnected iff for A being Subset of X st A is open holds Cl A = Int Cl A; theorem :: TDLAT_3:33 X is extremally_disconnected iff for A being Subset of X st A is closed holds Int A = Cl Int A; theorem :: TDLAT_3:34 X is extremally_disconnected iff for A being Subset of X st A is condensed holds A is closed & A is open; theorem :: TDLAT_3:35 X is extremally_disconnected iff for A being Subset of X st A is condensed holds A is closed_condensed & A is open_condensed; theorem :: TDLAT_3:36 X is extremally_disconnected iff for A being Subset of X st A is condensed holds Int Cl A = Cl Int A; theorem :: TDLAT_3:37 X is extremally_disconnected iff for A being Subset of X st A is condensed holds Int A = Cl A; theorem :: TDLAT_3:38 X is extremally_disconnected iff for A being Subset of X holds (A is open_condensed implies A is closed_condensed) & (A is closed_condensed implies A is open_condensed); definition let IT be non empty TopSpace; attr IT is hereditarily_extremally_disconnected means :: TDLAT_3:def 5 for X0 being non empty SubSpace of IT holds X0 is extremally_disconnected; end; definition cluster hereditarily_extremally_disconnected strict (non empty TopSpace); end; definition cluster hereditarily_extremally_disconnected -> extremally_disconnected (non empty TopSpace); cluster almost_discrete -> hereditarily_extremally_disconnected (non empty TopSpace); end; theorem :: TDLAT_3:39 for X being extremally_disconnected (non empty TopSpace), X0 being non empty SubSpace of X, A being Subset of X st A = the carrier of X0 & A is dense holds X0 is extremally_disconnected; definition let X be extremally_disconnected (non empty TopSpace); cluster open -> extremally_disconnected (non empty SubSpace of X); end; definition let X be extremally_disconnected (non empty TopSpace); cluster extremally_disconnected strict (non empty SubSpace of X); end; definition let X be hereditarily_extremally_disconnected (non empty TopSpace); cluster -> hereditarily_extremally_disconnected (non empty SubSpace of X); end; definition let X be hereditarily_extremally_disconnected (non empty TopSpace); cluster hereditarily_extremally_disconnected strict (non empty SubSpace of X); end; theorem :: TDLAT_3:40 (for X0 being closed non empty SubSpace of X holds X0 is extremally_disconnected) implies X is hereditarily_extremally_disconnected; begin :: 5. The Lattice of Domains of Extremally Disconnected Spaces. ::Properties of the Lattice of Domains of an Extremally Disconnected Space. reserve Y for extremally_disconnected (non empty TopSpace); theorem :: TDLAT_3:41 Domains_of Y = Closed_Domains_of Y; theorem :: TDLAT_3:42 D-Union Y = CLD-Union Y & D-Meet Y = CLD-Meet Y; theorem :: TDLAT_3:43 Domains_Lattice Y = Closed_Domains_Lattice Y; theorem :: TDLAT_3:44 Domains_of Y = Open_Domains_of Y; theorem :: TDLAT_3:45 D-Union Y = OPD-Union Y & D-Meet Y = OPD-Meet Y; theorem :: TDLAT_3:46 Domains_Lattice Y = Open_Domains_Lattice Y; theorem :: TDLAT_3:47 for A, B being Element of Domains_of Y holds (D-Union Y).(A,B) = A \/ B & (D-Meet Y).(A,B) = A /\ B; theorem :: TDLAT_3:48 for a, b being Element of Domains_Lattice Y for A, B being Element of Domains_of Y st a = A & b = B holds a "\/" b = A \/ B & a "/\" b = A /\ B; theorem :: TDLAT_3:49 for F being Subset-Family of Y st F is domains-family for S being Subset of Domains_Lattice Y st S = F holds "\/"(S,Domains_Lattice Y) = Cl(union F); theorem :: TDLAT_3:50 for F being Subset-Family of Y st F is domains-family for S being Subset of Domains_Lattice Y st S = F holds (S <> {} implies "/\"(S,Domains_Lattice Y) = Int(meet F)) & (S = {} implies "/\"(S,Domains_Lattice Y) = [#]Y); ::Lattice-theoretic Characterizations of Extremally Disconnected Spaces. reserve X for non empty TopSpace; theorem :: TDLAT_3:51 X is extremally_disconnected iff Domains_Lattice X is M_Lattice; theorem :: TDLAT_3:52 Domains_Lattice X = Closed_Domains_Lattice X implies X is extremally_disconnected; theorem :: TDLAT_3:53 Domains_Lattice X = Open_Domains_Lattice X implies X is extremally_disconnected; theorem :: TDLAT_3:54 Closed_Domains_Lattice X = Open_Domains_Lattice X implies X is extremally_disconnected; theorem :: TDLAT_3:55 X is extremally_disconnected iff Domains_Lattice X is B_Lattice;

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