:: Introduction to Meet-Continuous Topological Lattices
:: by Artur Korni{\l}owicz
::
:: Received November 3, 1998
:: Copyright (c) 1998-2021 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 STRUCT_0, XBOOLE_0, ZFMISC_1, ORDERS_2, WAYBEL_9, PRE_TOPC,
FINSET_1, SUBSET_1, RCOMP_1, TARSKI, NATTRA_1, CARD_5, RELAT_2, FUNCT_1,
WAYBEL_0, RELAT_1, SEQM_3, XXREAL_0, YELLOW_2, YELLOW_1, CARD_FIL,
ORDINAL2, WELLORD1, WAYBEL_1, FUNCT_2, REWRITE1, LATTICES, YELLOW_0,
LATTICE3, TDLAT_3, RLVECT_3, SETFAM_1, YELLOW_8, TOPS_1, CONNSP_2,
PRELAMB, CANTOR_1, WAYBEL_2, YELLOW13, CARD_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, TOPS_2, FUNCT_1, RELSET_1,
BINOP_1, FUNCT_2, FINSET_1, SETFAM_1, DOMAIN_1, STRUCT_0, ORDERS_2,
PRE_TOPC, TOPS_1, COMPTS_1, TDLAT_2, TDLAT_3, LATTICE3, CANTOR_1,
CONNSP_2, BORSUK_1, YELLOW_0, WAYBEL_0, YELLOW_1, ORDERS_3, WAYBEL_1,
YELLOW_2, YELLOW_3, WAYBEL_2, YELLOW_8, WAYBEL_9;
constructors SETFAM_1, TOPS_1, BORSUK_1, URYSOHN1, LATTICE3, TDLAT_2, TDLAT_3,
T_0TOPSP, CANTOR_1, ORDERS_3, WAYBEL_1, WAYBEL_9, YELLOW_8, COMPTS_1,
TOPS_2, WAYBEL_2;
registrations XBOOLE_0, SUBSET_1, RELSET_1, FINSET_1, STRUCT_0, PRE_TOPC,
TOPS_1, BORSUK_1, LATTICE3, YELLOW_0, TDLAT_3, TEX_1, WAYBEL_0, YELLOW_1,
YELLOW_2, WAYBEL_1, WAYBEL_3, YELLOW_9, YELLOW11, YELLOW12, RELAT_1;
requirements SUBSET, BOOLE;
begin :: Preliminaries
theorem :: YELLOW13:1
for T being T_1 non empty TopSpace, A being finite Subset of T
holds A is closed;
registration
let T be T_1 non empty TopSpace;
cluster finite -> closed for Subset of T;
end;
registration
let T be compact TopStruct;
cluster [#]T -> compact;
end;
registration
cluster finite T_1 -> discrete for non empty TopSpace;
end;
registration
cluster finite -> compact for TopSpace;
end;
theorem :: YELLOW13:2
for T being discrete non empty TopSpace holds T is normal;
theorem :: YELLOW13:3
for T being discrete non empty TopSpace holds T is regular;
theorem :: YELLOW13:4
for T being discrete non empty TopSpace holds T is T_2;
theorem :: YELLOW13:5
for T being discrete non empty TopSpace holds T is T_1;
registration
cluster discrete non empty -> normal regular T_2 T_1 for TopSpace;
end;
registration
cluster T_4 -> regular for non empty TopSpace;
end;
registration
cluster T_3 -> T_2 for TopSpace;
end;
theorem :: YELLOW13:6
for S being reflexive RelStr, T being reflexive transitive RelStr
, f being Function of S, T, X being Subset of S holds downarrow (f.:X) c=
downarrow (f.:downarrow X);
theorem :: YELLOW13:7
for S being reflexive RelStr, T being reflexive transitive RelStr, f
being Function of S, T, X being Subset of S st f is monotone holds downarrow (f
.:X) = downarrow (f.:downarrow X);
theorem :: YELLOW13:8
for N being non empty Poset holds IdsMap N is one-to-one;
registration
let N be non empty Poset;
cluster IdsMap N -> one-to-one;
end;
theorem :: YELLOW13:9
for N being finite LATTICE holds SupMap N is one-to-one;
registration
let N be finite LATTICE;
cluster SupMap N -> one-to-one;
end;
theorem :: YELLOW13:10
for N being finite LATTICE holds N, InclPoset Ids N are_isomorphic;
theorem :: YELLOW13:11
for N being complete non empty Poset, x being Element of N, X
being non empty Subset of N holds x"/\" preserves_inf_of X;
theorem :: YELLOW13:12
for N being complete non empty Poset, x being Element of N
holds x"/\" is meet-preserving;
registration
let N be complete non empty Poset, x be Element of N;
cluster x"/\" -> meet-preserving;
end;
begin :: On the basis of topological spaces
theorem :: YELLOW13:13
for T being anti-discrete non empty TopStruct, p being Point of T
holds {the carrier of T} is Basis of p;
theorem :: YELLOW13:14
for T being anti-discrete non empty TopStruct, p being Point of T, D
being Basis of p holds D = {the carrier of T};
theorem :: YELLOW13:15
for T being non empty TopSpace, P being Basis of T, p being Point of T
holds {A where A is Subset of T: A in P & p in A} is Basis of p;
theorem :: YELLOW13:16
for T being non empty TopStruct, A being Subset of T, p being Point of
T holds p in Cl A iff for K being Basis of p, Q being Subset of T st Q in K
holds A meets Q;
theorem :: YELLOW13:17
for T being non empty TopStruct, A being Subset of T, p being Point of
T holds p in Cl A iff ex K being Basis of p st for Q being Subset of T st Q in
K holds A meets Q;
definition
let T be TopStruct, p be Point of T;
mode basis of p -> Subset-Family of T means
:: YELLOW13:def 1
for A being Subset of T
st p in Int A ex P being Subset of T st P in it & p in Int P & P c= A;
end;
definition
let T be non empty TopSpace, p be Point of T;
redefine mode basis of p means
:: YELLOW13:def 2
for A being a_neighborhood of p ex P being
a_neighborhood of p st P in it & P c= A;
end;
theorem :: YELLOW13:18
for T being TopStruct, p being Point of T holds bool the carrier
of T is basis of p;
theorem :: YELLOW13:19
for T being non empty TopSpace, p being Point of T, P being
basis of p holds P is non empty;
registration
let T be non empty TopSpace, p be Point of T;
cluster -> non empty for basis of p;
end;
registration
let T be TopStruct, p be Point of T;
cluster non empty for basis of p;
end;
definition
let T be TopStruct, p be Point of T, P be basis of p;
attr P is correct means
:: YELLOW13:def 3
for A being Subset of T holds A in P iff p in Int A;
end;
registration
let T be TopStruct, p be Point of T;
cluster correct for basis of p;
end;
theorem :: YELLOW13:20
for T being TopStruct, p being Point of T holds {A where A is Subset
of T: p in Int A} is correct basis of p;
registration
let T be non empty TopSpace, p be Point of T;
cluster non empty correct for basis of p;
end;
theorem :: YELLOW13:21
for T being anti-discrete non empty TopStruct, p being Point of T
holds {the carrier of T} is correct basis of p;
theorem :: YELLOW13:22
for T being anti-discrete non empty TopStruct, p being Point of T, D
being correct basis of p holds D = {the carrier of T};
theorem :: YELLOW13:23
for T being non empty TopSpace, p being Point of T, P being Basis of p
holds P is basis of p;
definition
let T be TopStruct;
mode basis of T -> Subset-Family of T means
:: YELLOW13:def 4
for p being Point of T holds it is basis of p;
end;
theorem :: YELLOW13:24
for T being TopStruct holds bool the carrier of T is basis of T;
theorem :: YELLOW13:25
for T being non empty TopSpace, P being basis of T holds P is non empty;
registration
let T be non empty TopSpace;
cluster -> non empty for basis of T;
end;
registration
let T be TopStruct;
cluster non empty for basis of T;
end;
theorem :: YELLOW13:26
for T being non empty TopSpace, P being basis of T holds the topology
of T c= UniCl Int P;
theorem :: YELLOW13:27
for T being TopSpace, P being Basis of T holds P is basis of T;
definition
let T be non empty TopSpace-like TopRelStr;
attr T is topological_semilattice means
:: YELLOW13:def 5
for f being Function of [:T,T qua TopSpace:], T st f = inf_op T
holds f is continuous;
end;
registration
cluster reflexive -> topological_semilattice
for 1-element TopSpace-like TopRelStr;
end;
theorem :: YELLOW13:28
for T being topological_semilattice non empty TopSpace-like
TopRelStr, x being Element of T holds x"/\" is continuous;
registration
let T be topological_semilatticenon empty TopSpace-like TopRelStr, x be
Element of T;
cluster x"/\" -> continuous;
end;