Foundations of Vietoris homology theory with applications to non-compact spaces

Robert E. Reed. Foundations of Vietoris homology theory with applications to non-compact spaces. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1980. <http://eudml.org/doc/268548>.

@book{RobertE1980,
abstract = {CONTENTSPreface...................................................................................................................................... 5I. Introduction............................................................................................................................ 7II. Simple chains 2.1. Simplexes............................................................................................................ 12 2.2. Chains........................................................................................................................... 13 2.3. Boundary operator. Cycles and boundaries.......................................................... 15 2.4. Join operator................................................................................................................ 15 2.5. ε-simplexes and ε-chains........................................................................................... 16III. Sequential chains 3.1. Sequences and subsequences...................................................................... 17 3.2. Sequential chains....................................................................................................... 18 3.3. Infinite chains. General homology groups............................................................. 18 3.4. Infinite chains in subspaces..................................................................................... 19 3.5. True cycles. Vietoris homology groups................................................................... 21 3.6. Subsequences of infinite chains............................................................................. 22 3.7. A condition for homology of infinite cycles.............................................................. 24IV. Functions, mappings, and null translations4.1. Homomorphisms of simple chains induced by functions...................................... 25 4.2. Homomorphisms of sequential chains induced by functions........................... 25 4.3. Homomorphisms of ε-chains induced by functions............................................ 26 4.4. Homomorphisms of infinite chains induced by maps........................................ 27 4.5. Topological invariance of the central and Vietoris homology groups............... 28 4.6. Non-equivalence of the general and Vietoris homology groups....................... 30 4.7. The homotopy theorem.............................................................................................. 31 4.8. Null translations.......................................................................................................... 34V. The Phragmen-Brouwer theorem 5.1. Introduction.......................................................................................................... 37 5.2. The Phragmen Brouwer theorem for non-compact spaces............................... 39VI. The Alexandroff dimension theorem 6.1. Introduction.......................................................................................................... 40 6.2. Compactly dimensioned spaces............................................................................. 41 6.3. The generalized Alexandroff theorem..................................................................... 43Bibliography.............................................................................................................................. 46},
author = {Robert E. Reed},
keywords = {Vietoris homology theory with compact carriers in arbitrary metric spaces; extension of Alexandroff's homological dimension theorem to a class of locally compact metric spaces; Phragmen-Brouwer theorem for arbitrary metric spaces; covering dimension; compactly dimensioned spaces},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Foundations of Vietoris homology theory with applications to non-compact spaces},
url = {http://eudml.org/doc/268548},
year = {1980},
}

TY - BOOK
AU - Robert E. Reed
TI - Foundations of Vietoris homology theory with applications to non-compact spaces
PY - 1980
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - CONTENTSPreface...................................................................................................................................... 5I. Introduction............................................................................................................................ 7II. Simple chains 2.1. Simplexes............................................................................................................ 12 2.2. Chains........................................................................................................................... 13 2.3. Boundary operator. Cycles and boundaries.......................................................... 15 2.4. Join operator................................................................................................................ 15 2.5. ε-simplexes and ε-chains........................................................................................... 16III. Sequential chains 3.1. Sequences and subsequences...................................................................... 17 3.2. Sequential chains....................................................................................................... 18 3.3. Infinite chains. General homology groups............................................................. 18 3.4. Infinite chains in subspaces..................................................................................... 19 3.5. True cycles. Vietoris homology groups................................................................... 21 3.6. Subsequences of infinite chains............................................................................. 22 3.7. A condition for homology of infinite cycles.............................................................. 24IV. Functions, mappings, and null translations4.1. Homomorphisms of simple chains induced by functions...................................... 25 4.2. Homomorphisms of sequential chains induced by functions........................... 25 4.3. Homomorphisms of ε-chains induced by functions............................................ 26 4.4. Homomorphisms of infinite chains induced by maps........................................ 27 4.5. Topological invariance of the central and Vietoris homology groups............... 28 4.6. Non-equivalence of the general and Vietoris homology groups....................... 30 4.7. The homotopy theorem.............................................................................................. 31 4.8. Null translations.......................................................................................................... 34V. The Phragmen-Brouwer theorem 5.1. Introduction.......................................................................................................... 37 5.2. The Phragmen Brouwer theorem for non-compact spaces............................... 39VI. The Alexandroff dimension theorem 6.1. Introduction.......................................................................................................... 40 6.2. Compactly dimensioned spaces............................................................................. 41 6.3. The generalized Alexandroff theorem..................................................................... 43Bibliography.............................................................................................................................. 46
LA - eng
KW - Vietoris homology theory with compact carriers in arbitrary metric spaces; extension of Alexandroff's homological dimension theorem to a class of locally compact metric spaces; Phragmen-Brouwer theorem for arbitrary metric spaces; covering dimension; compactly dimensioned spaces
UR - http://eudml.org/doc/268548
ER -