Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Some Properties of Isomorphism between Relational Structures. On the Product of Topological Spaces

**by****Jaroslaw Gryko, and****Artur Kornilowicz**- Received June 22, 1999
- MML identifier: YELLOW14

- [ Mizar article, MML identifier index ]

environ vocabulary RELAT_1, FUNCT_1, BOOLE, FUNCT_4, CAT_1, ORDERS_1, PRE_TOPC, SUBSET_1, RELAT_2, WAYBEL_0, QUANTAL1, PRALG_1, MONOID_0, FUNCOP_1, SEQM_3, WAYBEL_9, WAYBEL24, YELLOW_0, GROUP_1, CARD_3, SGRAPH1, WELLORD1, LATTICE3, LATTICES, T_0TOPSP, TOPS_2, ORDINAL2, WAYBEL11, WAYBEL_6, YELLOW_8, COMPTS_1, SETFAM_1, FINSET_1, TARSKI, WAYBEL_3, TOPS_1, WAYBEL18, PBOOLE, RLVECT_2, BORSUK_1, CANTOR_1, WAYBEL_1, FUNCT_3, GROUP_6, TSP_1, URYSOHN1, SEQ_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, RELSET_1, FINSET_1, TOPS_2, FUNCT_1, PARTFUN1, FUNCT_4, CQC_LANG, FUNCT_7, CARD_3, MONOID_0, PBOOLE, PRALG_1, PRE_CIRC, STRUCT_0, PRE_TOPC, GRCAT_1, TOPS_1, COMPTS_1, T_0TOPSP, TSP_1, URYSOHN1, CANTOR_1, TMAP_1, ORDERS_1, LATTICE3, WAYBEL18, YELLOW_0, WAYBEL_0, WAYBEL_1, FUNCT_2, YELLOW_1, YELLOW_2, WAYBEL_3, YELLOW_6, YELLOW_8, WAYBEL_9, WAYBEL11, WAYBEL24; constructors BORSUK_3, CANTOR_1, CQC_LANG, FUNCT_7, GRCAT_1, ORDERS_3, PRE_CIRC, TMAP_1, TOLER_1, TOPS_1, TOPS_2, URYSOHN1, WAYBEL_1, WAYBEL17, WAYBEL18, WAYBEL24, YELLOW_8, MONOID_0; clusters BORSUK_3, FUNCT_1, TEX_4, PRE_TOPC, RELAT_1, YELLOW_8, CQC_LANG, RELSET_1, STRUCT_0, TEX_1, TSP_1, TOPGRP_1, YELLOW_0, YELLOW_1, YELLOW_6, YELLOW12, WAYBEL_0, WAYBEL_3, WAYBEL_7, WAYBEL12, WAYBEL18, WAYBEL19, WAYBEL24, SUBSET_1, PARTFUN1, FUNCT_2; requirements SUBSET, BOOLE; begin :: Preliminaries theorem :: YELLOW14:1 bool 1 = {0,1}; theorem :: YELLOW14:2 for X being set, Y being Subset of X holds rng ((id X)|Y) = Y; theorem :: YELLOW14:3 for f being Function, a, b being set holds (f +* (a .--> b)).a = b; definition cluster strict empty RelStr; end; theorem :: YELLOW14:4 for S being empty 1-sorted, T being 1-sorted, f being map of S, T st rng f = [#]T holds T is empty; theorem :: YELLOW14:5 for S being 1-sorted, T being empty 1-sorted, f being map of S, T st dom f = [#]S holds S is empty; theorem :: YELLOW14:6 for S being non empty 1-sorted, T being 1-sorted, f being map of S, T st dom f = [#]S holds T is non empty; theorem :: YELLOW14:7 for S being 1-sorted, T being non empty 1-sorted, f being map of S, T st rng f = [#]T holds S is non empty; definition let S be non empty reflexive RelStr, T be non empty RelStr, f be map of S, T; redefine attr f is directed-sups-preserving means :: YELLOW14:def 1 for X being non empty directed Subset of S holds f preserves_sup_of X; end; definition let R be 1-sorted, N be NetStr over R; attr N is Function-yielding means :: YELLOW14:def 2 the mapping of N is Function-yielding; end; definition cluster strict non empty constituted-Functions 1-sorted; end; definition cluster strict non empty constituted-Functions RelStr; end; definition let R be constituted-Functions 1-sorted; cluster -> Function-yielding NetStr over R; end; definition let R be constituted-Functions 1-sorted; cluster strict Function-yielding NetStr over R; end; definition let R be non empty constituted-Functions 1-sorted; cluster strict non empty Function-yielding NetStr over R; end; definition let R be constituted-Functions 1-sorted, N be Function-yielding NetStr over R; cluster the mapping of N -> Function-yielding; end; definition let R be non empty constituted-Functions 1-sorted; cluster strict Function-yielding net of R; end; definition let S be non empty 1-sorted, N be non empty NetStr over S; cluster rng the mapping of N -> non empty; end; definition let S be non empty 1-sorted, N be non empty NetStr over S; cluster rng netmap (N,S) -> non empty; end; theorem :: YELLOW14:8 for A, B, C being non empty RelStr, f being map of B, C, g, h being map of A, B st g <= h & f is monotone holds f * g <= f * h; theorem :: YELLOW14:9 for S being non empty TopSpace, T being non empty TopSpace-like TopRelStr, f, g being map of S, T, x, y being Element of ContMaps(S,T) st x = f & y = g holds x <= y iff f <= g; definition let I be set, R be non empty RelStr; cluster -> Function-like Relation-like Element of R |^ I; end; definition let I be non empty set, R be non empty RelStr, f be Element of R |^ I, i be Element of I; redefine func f.i -> Element of R; end; begin :: Some properties of isomorphism between relational structures theorem :: YELLOW14:10 for S, T being RelStr, f being map of S, T st f is isomorphic holds f is onto; definition let S, T be RelStr; cluster isomorphic -> onto map of S, T; end; theorem :: YELLOW14:11 for S, T being non empty RelStr, f being map of S, T st f is isomorphic holds f/" is isomorphic; theorem :: YELLOW14:12 for S, T being non empty RelStr st S, T are_isomorphic & S is with_infima holds T is with_infima; theorem :: YELLOW14:13 for S, T being non empty RelStr st S, T are_isomorphic & S is with_suprema holds T is with_suprema; theorem :: YELLOW14:14 for L being RelStr st L is empty holds L is bounded; definition cluster empty -> bounded RelStr; end; theorem :: YELLOW14:15 for S, T being RelStr st S, T are_isomorphic & S is lower-bounded holds T is lower-bounded; theorem :: YELLOW14:16 for S, T being RelStr st S, T are_isomorphic & S is upper-bounded holds T is upper-bounded; theorem :: YELLOW14:17 for S, T being non empty RelStr, A being Subset of S, f being map of S, T st f is isomorphic & ex_sup_of A, S holds ex_sup_of f.:A, T; theorem :: YELLOW14:18 for S, T being non empty RelStr, A being Subset of S, f being map of S, T st f is isomorphic & ex_inf_of A, S holds ex_inf_of f.:A, T; begin :: On the product of topological spaces theorem :: YELLOW14:19 for S, T being TopStruct st (S,T are_homeomorphic or ex f being map of S,T st dom f = [#]S & rng f = [#]T) holds S is empty iff T is empty; theorem :: YELLOW14:20 for T being non empty TopSpace holds T, the TopStruct of T are_homeomorphic; definition let T be Scott (reflexive non empty TopRelStr); cluster open -> inaccessible upper Subset of T; cluster inaccessible upper -> open Subset of T; end; theorem :: YELLOW14:21 for T being TopStruct, x, y being Point of T, X, Y being Subset of T st X = {x} & Cl X c= Cl Y holds x in Cl Y; theorem :: YELLOW14:22 for T being TopStruct, x, y being Point of T, Y, V being Subset of T st Y = {y} & x in Cl Y & V is open & x in V holds y in V; theorem :: YELLOW14:23 for T being TopStruct, x, y being Point of T, X, Y being Subset of T st X = {x} & Y = {y} holds (for V being Subset of T st V is open holds (x in V implies y in V)) implies Cl X c= Cl Y; theorem :: YELLOW14:24 for S, T being non empty TopSpace, A being irreducible Subset of S, B being Subset of T st A = B & the TopStruct of S = the TopStruct of T holds B is irreducible; theorem :: YELLOW14:25 for S, T being non empty TopSpace, a being Point of S, b being Point of T, A being Subset of S, B being Subset of T st a = b & A = B & the TopStruct of S = the TopStruct of T & a is_dense_point_of A holds b is_dense_point_of B; theorem :: YELLOW14:26 for S, T being TopStruct, A being Subset of S, B being Subset of T st A = B & the TopStruct of S = the TopStruct of T & A is compact holds B is compact; theorem :: YELLOW14:27 for S, T being non empty TopSpace st the TopStruct of S = the TopStruct of T & S is sober holds T is sober; theorem :: YELLOW14:28 for S, T being non empty TopSpace st the TopStruct of S = the TopStruct of T & S is locally-compact holds T is locally-compact; theorem :: YELLOW14:29 for S, T being TopStruct st the TopStruct of S = the TopStruct of T & S is compact holds T is compact; definition let I be non empty set, T be non empty TopSpace, x be Point of product (I --> T), i be Element of I; redefine func x.i -> Element of T; end; theorem :: YELLOW14:30 for M being non empty set, J being TopSpace-yielding non-Empty ManySortedSet of M, x, y being Point of product J holds x in Cl {y} iff for i being Element of M holds x.i in Cl {y.i}; theorem :: YELLOW14:31 for M being non empty set, T being non empty TopSpace, x, y being Point of product (M --> T) holds x in Cl {y} iff for i being Element of M holds x.i in Cl {y.i}; theorem :: YELLOW14:32 for M being non empty set, i being Element of M, J being TopSpace-yielding non-Empty ManySortedSet of M, x being Point of product J holds pi(Cl {x}, i) = Cl {x.i}; theorem :: YELLOW14:33 for M being non empty set, i being Element of M, T being non empty TopSpace, x being Point of product (M --> T) holds pi(Cl {x}, i) = Cl {x.i}; theorem :: YELLOW14:34 for X, Y being non empty TopStruct, f being map of X, Y, g being map of Y, X st f = id X & g = id X & f is continuous & g is continuous holds the TopStruct of X = the TopStruct of Y; theorem :: YELLOW14:35 for X, Y being non empty TopSpace, f being map of X, Y st corestr f is continuous holds f is continuous; definition let X be non empty TopSpace, Y be non empty SubSpace of X; cluster incl Y -> continuous; end; theorem :: YELLOW14:36 for T being non empty TopSpace, f being map of T, T st f*f = f holds corestr f * incl Image f = id Image f; theorem :: YELLOW14:37 for Y being non empty TopSpace, W being non empty SubSpace of Y holds corestr incl W is_homeomorphism; theorem :: YELLOW14:38 for M being non empty set, J being TopSpace-yielding non-Empty ManySortedSet of M st for i being Element of M holds J.i is T_0 TopSpace holds product J is T_0; definition let I be non empty set, T be non empty T_0 TopSpace; cluster product (I --> T) -> T_0; end; theorem :: YELLOW14:39 for M being non empty set, J being TopSpace-yielding non-Empty ManySortedSet of M st for i being Element of M holds J.i is being_T1 TopSpace-like holds product J is_T1; definition let I be non empty set, T be non empty being_T1 TopSpace; cluster product (I --> T) -> being_T1; end;

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