Page 8 - Textos de Matemática Vol. 41
P. 8

vi Contents 4.5 Extremalresp.strongmonomorphismsinLoc . . . . . . . . . . . 21
4.6 AfactorizationsysteminLoc .................... 22
5 Sublocales, that is, generalized subspaces 25
5.1 One-onelocalicmaps......................... 25 5.2 Sublocales ............................... 25 5.3 Alternativerepresentationsofsublocales . . . . . . . . . . . . . . 27 5.4 Theco-frameofsublocales...................... 29 5.5 Subspacesinthiscontext....................... 30 5.6 Thecongruencelattice ........................ 30
6 Some special sublocales 31
6.1 One-pointsublocales ......................... 31
6.2 Openandclosedsublocales ..................... 31
6.3 Closureofasublocale......................... 34
6.4 Density................................. 34
6.5 Every sublocale is constructed from closed and open ones . . . . . 35
6.6 Booleansublocales .......................... 36
6.7 Complementedsublocales ...................... 37
6.8 Spatialsublocales........................... 38
7 Images and preimages 43
7.1 Images................................. 43 7.2 Closedmaps.............................. 44 7.3 Openmaps .............................. 45 7.4 ThesublocaleAsloc .......................... 47 7.5 Preimages ............................... 47 7.6 f−1[−]asacoframehomomorphism................. 48
8 Sublocales determined by binary relations 51
8.1 Thesublocalegeneratedbyarelation. . . . . . . . . . . . . . . . 51 8.2 Saturatedelements .......................... 52 8.3 FactorizationTheorem ........................ 53 8.4 A more transparent formula for saturatedness . . . . . . . . . . . 53 8.5 Inducedcongruences ......................... 54
9 Products. Completeness of the category Loc 57 9.1 Notation................................ 57 9.2 Down-sets ............................... 58 9.3 Theproduct i∈JLi ......................... 58 9.4 Ausefulsimpledistributiverule................... 60 9.5 Thediagonalmorphism ....................... 61 9.6 Completeness ............................. 62 9.7 Cocompleteness............................ 62

















































































   6   7   8   9   10