Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Injective Spaces. Part II

**by****Artur Kornilowicz, and****Jaroslaw Gryko**- Received July 3, 1999
- MML identifier: WAYBEL25

- [ Mizar article, MML identifier index ]

environ vocabulary PRE_TOPC, WAYBEL18, BOOLE, SUBSET_1, URYSOHN1, BHSP_3, WAYBEL11, METRIC_1, WAYBEL_9, FUNCTOR0, T_0TOPSP, YELLOW_9, CARD_3, FUNCOP_1, YELLOW_1, RELAT_1, RLVECT_2, FUNCT_1, ORDERS_1, WAYBEL_0, CAT_1, TOPS_2, ORDINAL2, WELLORD1, GROUP_6, FUNCT_3, WAYBEL_1, BORSUK_1, LATTICES, QUANTAL1, LATTICE3, YELLOW_0, WAYBEL_3, RELAT_2, PROB_1, BINOP_1, SEQM_3, WAYBEL24, FUNCT_2, GROUP_1, PRALG_2, PRALG_1, YELLOW_2, REALSET1, CONNSP_2, TOPS_1, SEQ_2, COMPTS_1, YELLOW_8, TARSKI, SETFAM_1, WAYBEL25; notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, RELAT_1, RELSET_1, FUNCT_1, PARTFUN1, FUNCT_2, REALSET1, NATTRA_1, TOLER_1, QUANTAL1, CARD_3, PRALG_1, PRALG_2, PRE_CIRC, WAYBEL18, STRUCT_0, PRE_TOPC, GRCAT_1, TOPS_1, TOPS_2, COMPTS_1, BORSUK_1, TMAP_1, T_0TOPSP, URYSOHN1, BORSUK_3, ORDERS_1, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_2, WAYBEL_1, WAYBEL_2, YELLOW_6, WAYBEL_3, YELLOW_8, WAYBEL_9, WAYBEL11, YELLOW_9, WAYBEL17, YELLOW_1, WAYBEL24, YELLOW14; constructors BORSUK_3, CANTOR_1, ENUMSET1, GRCAT_1, NATTRA_1, ORDERS_3, PRALG_2, QUANTAL1, RELAT_2, TMAP_1, TOLER_1, TOPS_1, TOPS_2, URYSOHN1, WAYBEL_1, WAYBEL_8, WAYBEL17, WAYBEL24, YELLOW_8, YELLOW_9, YELLOW14, MONOID_0, MEMBERED; clusters BORSUK_3, FUNCT_1, LATTICE3, PRALG_1, PRE_TOPC, RELSET_1, STRUCT_0, TEX_1, TSP_1, TOPS_1, YELLOW_0, YELLOW_1, YELLOW_9, YELLOW12, WAYBEL_0, WAYBEL_2, WAYBEL_8, WAYBEL10, WAYBEL12, WAYBEL17, WAYBEL18, WAYBEL19, WAYBEL21, WAYBEL24, YELLOW14, SUBSET_1, BORSUK_1, MEMBERED, ZFMISC_1, FUNCT_2; requirements SUBSET, BOOLE; begin :: Injective spaces theorem :: WAYBEL25:1 for p being Point of Sierpinski_Space st p = 0 holds {p} is closed; theorem :: WAYBEL25:2 for p being Point of Sierpinski_Space st p = 1 holds {p} is non closed; definition cluster Sierpinski_Space -> non being_T1; end; definition cluster complete Scott -> discerning TopLattice; end; definition cluster injective strict T_0-TopSpace; end; definition cluster complete Scott strict TopLattice; end; theorem :: WAYBEL25:3 :: see WAYBEL18:16 for I being non empty set, T being Scott TopAugmentation of product(I --> BoolePoset 1) holds the carrier of T = the carrier of product(I --> Sierpinski_Space); theorem :: WAYBEL25:4 for L1, L2 being complete LATTICE, T1 being Scott TopAugmentation of L1, T2 being Scott TopAugmentation of L2, h being map of L1, L2, H being map of T1, T2 st h = H & h is isomorphic holds H is_homeomorphism; theorem :: WAYBEL25:5 for L1, L2 being complete LATTICE, T1 being Scott TopAugmentation of L1, T2 being Scott TopAugmentation of L2 st L1, L2 are_isomorphic holds T1, T2 are_homeomorphic; theorem :: WAYBEL25:6 for S, T being non empty TopSpace st S is injective & S, T are_homeomorphic holds T is injective; theorem :: WAYBEL25:7 for L1, L2 being complete LATTICE, T1 being Scott TopAugmentation of L1, T2 being Scott TopAugmentation of L2 st L1, L2 are_isomorphic & T1 is injective holds T2 is injective; definition let X, Y be non empty TopSpace; redefine pred X is_Retract_of Y means :: WAYBEL25:def 1 ex c being continuous map of X, Y, r being continuous map of Y, X st r * c = id X; end; theorem :: WAYBEL25:8 for S, T, X, Y being non empty TopSpace st the TopStruct of S = the TopStruct of T & the TopStruct of X = the TopStruct of Y & S is_Retract_of X holds T is_Retract_of Y; theorem :: WAYBEL25:9 for T, S1, S2 being non empty TopStruct st S1, S2 are_homeomorphic & S1 is_Retract_of T holds S2 is_Retract_of T; theorem :: WAYBEL25:10 for S, T being non empty TopSpace st T is injective & S is_Retract_of T holds S is injective; ::p.126 exercise 3.13, 1 => 2 theorem :: WAYBEL25:11 for J being injective non empty TopSpace, Y being non empty TopSpace st J is SubSpace of Y holds J is_Retract_of Y; :: p.123 proposition 3.5 :: p.124 theorem 3.8 (i, part a) :: p.126 exercise 3.14 theorem :: WAYBEL25:12 for L being complete continuous LATTICE, T being Scott TopAugmentation of L holds T is injective; definition let L be complete continuous LATTICE; cluster Scott -> injective TopAugmentation of L; end; definition let T be injective non empty TopSpace; cluster the TopStruct of T -> injective; end; begin :: Specialization order ::p.124 definition 3.6 definition let T be TopStruct; func Omega T -> strict TopRelStr means :: WAYBEL25:def 2 the TopStruct of it = the TopStruct of T & for x, y being Element of it holds x <= y iff ex Y being Subset of T st Y = {y} & x in Cl Y; end; definition let T be empty TopStruct; cluster Omega T -> empty; end; definition let T be non empty TopStruct; cluster Omega T -> non empty; end; definition let T be TopSpace; cluster Omega T -> TopSpace-like; end; definition let T be TopStruct; cluster Omega T -> reflexive; end; definition let T be TopStruct; cluster Omega T -> transitive; end; definition let T be T_0-TopSpace; cluster Omega T -> antisymmetric; end; theorem :: WAYBEL25:13 for S, T being TopSpace st the TopStruct of S = the TopStruct of T holds Omega S = Omega T; theorem :: WAYBEL25:14 for M being non empty set, T being non empty TopSpace holds the RelStr of Omega product(M --> T) = the RelStr of product(M --> Omega T); ::p.124 theorem 3.8 (i, part b) theorem :: WAYBEL25:15 for S being Scott complete TopLattice holds Omega S = the TopRelStr of S; ::p.124 theorem 3.8 (i, part b) theorem :: WAYBEL25:16 for L being complete LATTICE, S being Scott TopAugmentation of L holds the RelStr of Omega S = the RelStr of L; definition let S be Scott complete TopLattice; cluster Omega S -> complete; end; theorem :: WAYBEL25:17 for T being non empty TopSpace, S being non empty SubSpace of T holds Omega S is full SubRelStr of Omega T; theorem :: WAYBEL25:18 for S, T being TopSpace, h being map of S, T, g being map of Omega S, Omega T st h = g & h is_homeomorphism holds g is isomorphic; theorem :: WAYBEL25:19 for S, T being TopSpace st S, T are_homeomorphic holds Omega S, Omega T are_isomorphic; ::p.124 proposition 3.7 ::p.124 theorem 3.8 (ii, part a) theorem :: WAYBEL25:20 for T being injective T_0-TopSpace holds Omega T is complete continuous LATTICE; definition let T be injective T_0-TopSpace; cluster Omega T -> complete continuous; end; theorem :: WAYBEL25:21 for X, Y being non empty TopSpace, f being continuous map of Omega X, Omega Y holds f is monotone; definition let X, Y be non empty TopSpace; cluster continuous -> monotone map of Omega X, Omega Y; end; theorem :: WAYBEL25:22 for T being non empty TopSpace, x being Element of Omega T holds Cl {x} = downarrow x; definition let T be non empty TopSpace, x be Element of Omega T; cluster Cl {x} -> non empty lower directed; cluster downarrow x -> closed; end; theorem :: WAYBEL25:23 for X being TopSpace, A being open Subset of Omega X holds A is upper; definition let T be TopSpace; cluster open -> upper Subset of Omega T; end; definition let I be non empty set, S, T be non empty RelStr, N be net of T, i be Element of I such that the carrier of T c= the carrier of S |^ I; func commute(N,i,S) -> strict net of S means :: WAYBEL25:def 3 the RelStr of it = the RelStr of N & the mapping of it = (commute the mapping of N).i; end; theorem :: WAYBEL25:24 for X, Y being non empty TopSpace, N being net of ContMaps(X,Omega Y), i being Element of N, x being Point of X holds (the mapping of commute(N,x,Omega Y)).i = (the mapping of N).i.x; theorem :: WAYBEL25:25 for X, Y being non empty TopSpace, N being eventually-directed net of ContMaps(X,Omega Y), x being Point of X holds commute(N,x,Omega Y) is eventually-directed; definition let X, Y be non empty TopSpace, N be eventually-directed net of ContMaps(X,Omega Y), x be Point of X; cluster commute(N,x,Omega Y) -> eventually-directed; end; definition let X, Y be non empty TopSpace; cluster -> Function-yielding net of ContMaps(X,Omega Y); end; theorem :: WAYBEL25:26 for X being non empty TopSpace, Y being T_0-TopSpace, N being net of ContMaps(X,Omega Y) st for x being Point of X holds ex_sup_of commute(N,x,Omega Y) holds ex_sup_of rng the mapping of N, (Omega Y) |^ the carrier of X; begin :: Monotone convergence topological spaces ::p.125 definition 3.9 definition let T be non empty TopSpace; attr T is monotone-convergence means :: WAYBEL25:def 4 for D being non empty directed Subset of Omega T holds ex_sup_of D,Omega T & for V being open Subset of T st sup D in V holds D meets V; end; theorem :: WAYBEL25:27 for S, T being non empty TopSpace st the TopStruct of S = the TopStruct of T & S is monotone-convergence holds T is monotone-convergence; definition cluster trivial -> monotone-convergence T_0-TopSpace; end; definition cluster strict trivial non empty TopSpace; end; theorem :: WAYBEL25:28 for S being monotone-convergence T_0-TopSpace, T being T_0-TopSpace st S, T are_homeomorphic holds T is monotone-convergence; theorem :: WAYBEL25:29 for S being Scott complete TopLattice holds S is monotone-convergence; definition let L be complete LATTICE; cluster -> monotone-convergence (Scott TopAugmentation of L); end; definition let L be complete LATTICE, S be Scott TopAugmentation of L; cluster the TopStruct of S -> monotone-convergence; end; theorem :: WAYBEL25:30 for X being monotone-convergence T_0-TopSpace holds Omega X is up-complete; definition let X be monotone-convergence T_0-TopSpace; cluster Omega X -> up-complete; end; theorem :: WAYBEL25:31 for X being monotone-convergence (non empty TopSpace), N being eventually-directed prenet of Omega X holds ex_sup_of N; theorem :: WAYBEL25:32 for X being monotone-convergence (non empty TopSpace), N being eventually-directed net of Omega X holds sup N in Lim N; theorem :: WAYBEL25:33 for X being monotone-convergence (non empty TopSpace), N being eventually-directed net of Omega X holds N is convergent; definition let X be monotone-convergence (non empty TopSpace); cluster -> convergent (eventually-directed net of Omega X); end; theorem :: WAYBEL25:34 for X being non empty TopSpace st for N being eventually-directed net of Omega X holds ex_sup_of N & sup N in Lim N holds X is monotone-convergence; ::p.125 lemma 3.10 theorem :: WAYBEL25:35 for X being monotone-convergence (non empty TopSpace), Y being T_0-TopSpace, f being continuous map of Omega X, Omega Y holds f is directed-sups-preserving; definition let X be monotone-convergence (non empty TopSpace), Y be T_0-TopSpace; cluster continuous -> directed-sups-preserving map of Omega X, Omega Y; end; theorem :: WAYBEL25:36 for T being monotone-convergence T_0-TopSpace, R being T_0-TopSpace st R is_Retract_of T holds R is monotone-convergence; ::p.124 theorem 3.8 (ii, part b) theorem :: WAYBEL25:37 for T being injective T_0-TopSpace, S being Scott TopAugmentation of Omega T holds the TopStruct of S = the TopStruct of T; theorem :: WAYBEL25:38 for T being injective T_0-TopSpace holds T is compact locally-compact sober; theorem :: WAYBEL25:39 for T being injective T_0-TopSpace holds T is monotone-convergence; definition cluster injective -> monotone-convergence T_0-TopSpace; end; theorem :: WAYBEL25:40 for X being non empty TopSpace, Y being monotone-convergence T_0-TopSpace, N being net of ContMaps(X,Omega Y), f, g being map of X, Omega Y st f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) & ex_sup_of rng the mapping of N, (Omega Y) |^ the carrier of X & g in rng the mapping of N holds g <= f; theorem :: WAYBEL25:41 for X being non empty TopSpace, Y being monotone-convergence T_0-TopSpace, N being net of ContMaps(X,Omega Y), x being Point of X, f being map of X, Omega Y st (for a being Point of X holds commute(N,a,Omega Y) is eventually-directed) & f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) holds f.x = sup commute(N,x,Omega Y); ::p.125 lemma 3.11 theorem :: WAYBEL25:42 for X being non empty TopSpace, Y being monotone-convergence T_0-TopSpace, N being net of ContMaps(X,Omega Y) st for x being Point of X holds commute(N,x,Omega Y) is eventually-directed holds "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) is continuous map of X, Y; ::p.126 lemma 3.12 (i) theorem :: WAYBEL25:43 for X being non empty TopSpace, Y being monotone-convergence T_0-TopSpace holds ContMaps(X,Omega Y) is directed-sups-inheriting SubRelStr of (Omega Y) |^ the carrier of X;

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