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

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Reed R. E.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Preface...................................................................................................................................... 5 I. Introduction............................................................................................................................ 7 II. 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........................................................................................... 16 III. 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.............................................................. 24 IV. Functions, mappings, and null translations 4.1. 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.......................................................................................................... 34 V. The Phragmen-Brouwer theorem 5.1. Introduction.......................................................................................................... 37 5.2. The Phragmen Brouwer theorem for non-compact spaces............................... 39 VI. The Alexandroff dimension theorem 6.1. Introduction.......................................................................................................... 40 6.2. Compactly dimensioned spaces............................................................................. 41 6.3. The generalized Alexandroff theorem..................................................................... 43 Bibliography.............................................................................................................................. 46

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1 Reed R. E.

1 1980

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