:: The Tichonov Theorem :: by Bart{\l}omiej Skorulski :: :: Received May 23, 2000 :: Copyright (c) 2000-2018 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 FUNCT_1, SUBSET_1, CARD_3, RELAT_1, XBOOLE_0, FUNCT_4, TARSKI, WAYBEL18, WAYBEL_3, PBOOLE, STRUCT_0, RLVECT_2, PRE_TOPC, RCOMP_1, SETFAM_1, FINSET_1, CANTOR_1, YELLOW_1, ZFMISC_1; notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, RELAT_1, FUNCT_1, RELSET_1, FUNCT_7, FINSET_1, PBOOLE, PRALG_1, CARD_3, PRE_TOPC, TOPS_2, COMPTS_1, CANTOR_1, YELLOW_1, WAYBEL_3, WAYBEL18; constructors SETFAM_1, FUNCT_7, TOPS_2, COMPTS_1, CANTOR_1, MONOID_0, WAYBEL18, RELSET_1, FUNCT_4; registrations SUBSET_1, RELSET_1, FINSET_1, CARD_3, STRUCT_0, PRE_TOPC, MONOID_0, YELLOW_1, YELLOW_6, WAYBEL18; requirements SUBSET, BOOLE; begin ::Some Properties of Products theorem :: YELLOW17:1 for F being Function, i, xi being set, Ai being Subset of F.i holds proj(F,i)"({xi}) meets proj(F,i)"Ai implies xi in Ai; theorem :: YELLOW17:2 for F,f being Function, i,xi being set st xi in F.i & f in product F holds f+*(i,xi) in product F; theorem :: YELLOW17:3 for F being Function, i being set st i in dom F holds rng proj(F,i) c= F.i & (product F <> {} implies rng proj(F,i) = F.i); theorem :: YELLOW17:4 for F being Function, i being set st i in dom F holds proj(F,i)"(F.i) = product F; theorem :: YELLOW17:5 for F,f being Function, i,xi being set st xi in F.i & i in dom F & f in product F holds f+*(i,xi) in proj(F,i)"({xi}); theorem :: YELLOW17:6 for F,f being Function, i1,i2,xi1 being set, Ai2 being Subset of F.i2 st xi1 in F.i1 & f in product F holds i1 <> i2 implies (f in proj(F,i2)" Ai2 iff f+*(i1,xi1) in proj(F,i2)"Ai2); theorem :: YELLOW17:7 for F being Function, i1,i2,xi1 being set, Ai2 being Subset of F. i2 st product F <> {} & xi1 in F.i1 & i1 in dom F & i2 in dom F & Ai2<>F.i2 holds proj(F,i1)"({xi1}) c= proj(F,i2)"Ai2 iff i1 = i2 & xi1 in Ai2; scheme :: YELLOW17:sch 1 ElProductEx { I()->non empty set, J()->TopStruct-yielding non-Empty ManySortedSet of I(), P[object,object] }: ex f being Element of product J() st for i being Element of I() holds P[f.i,i] provided for i being Element of I() ex x being Element of J().i st P[x,i]; theorem :: YELLOW17:8 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, f being Element of product J holds proj(J,i).f=f.i; theorem :: YELLOW17:9 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, xi being Element of J.i, Ai being Subset of J.i holds proj(J,i)"({xi}) meets proj(J,i)"Ai implies xi in Ai; theorem :: YELLOW17:10 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I holds proj(J,i)"[#](J.i) = [#] product J; theorem :: YELLOW17:11 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, xi being Element of J.i, f being Element of product J holds f+*(i,xi) in proj(J,i)"({xi}); theorem :: YELLOW17:12 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i1,i2 being Element of I, xi1 being Element of J.i1, Ai2 being Subset of J.i2 st Ai2<>[#](J.i2) holds proj(J,i1)"({xi1}) c= proj(J,i2)" Ai2 iff i1 = i2 & xi1 in Ai2; theorem :: YELLOW17:13 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i1,i2 being Element of I, xi1 being Element of J.i1, Ai2 being Subset of J.i2, f being Element of product J st i1<> i2 holds f in proj(J ,i2)"Ai2 iff f+*(i1,xi1) in proj(J,i2)"Ai2; begin theorem :: YELLOW17:14 for T being non empty TopStruct holds T is compact iff for F being Subset-Family of T st F is open & [#](T) c= union(F) ex G being Subset-Family of T st G c= F & [#]T c= union G & G is finite; theorem :: YELLOW17:15 for T being non empty TopSpace, B being prebasis of T holds T is compact iff for F being Subset of B st [#](T) c= union(F) ex G being finite Subset of F st [#]T c= union G; begin ::The Tichonov Theorem theorem :: YELLOW17:16 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, A being set st A in product_prebasis J ex i being Element of I, Ai being Subset of J.i st Ai is open & proj(J,i)"Ai = A; theorem :: YELLOW17:17 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, xi being Element of J.i, A being set st A in product_prebasis J & proj(J,i)"({xi}) c= A holds A = [#](product J) or ex Ai being Subset of J.i st Ai <> [#](J.i) & xi in Ai & Ai is open & A=proj(J, i)"Ai; theorem :: YELLOW17:18 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, Fi being non empty Subset-Family of J .i st [#](J.i) c= union(Fi) holds [#](product J) c= union the set of all proj(J,i)"Ai where Ai is Element of Fi; theorem :: YELLOW17:19 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, xi being Element of J.i, G being Subset of product_prebasis J st proj(J,i)"({xi}) c= union G & (for A being set st A in product_prebasis J & A in G holds not proj(J,i)"({xi}) c= A) holds [#]( product J) c= union G; theorem :: YELLOW17:20 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, F being Subset of product_prebasis J holds (for G being finite Subset of F holds not [#](product J) c= union G) implies for xi being Element of J.i, G being finite Subset of F st proj(J,i)"({ xi}) c= union G ex A being set st A in product_prebasis J & A in G & proj(J,i)" ({xi}) c= A; theorem :: YELLOW17:21 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, F being Subset of product_prebasis J holds (for G being finite Subset of F holds not [#](product J) c= union G) implies for xi being Element of J.i, G being finite Subset of F st proj(J,i)"({ xi}) c= union G holds ex Ai being Subset of J.i st Ai <> [#](J.i) & xi in Ai & proj(J,i)"Ai in G & Ai is open; theorem :: YELLOW17:22 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I, i being Element of I, F being Subset of product_prebasis J st (for i being Element of I holds J.i is compact) & (for G being finite Subset of F holds not [#](product J) c= union G) ex xi being Element of J.i st for G being finite Subset of F holds not proj(J,i)"({xi}) c= union G; ::\$N Tychonoff's theorem theorem :: YELLOW17:23 for I being non empty set, J being TopStruct-yielding non-Empty ManySortedSet of I st for i being Element of I holds J.i is compact holds product J is compact;