Strong homology and the proper forcing axiom

Authors:
Alan Dow, Petr Simon and Jerry E. Vaughan

Journal:
Proc. Amer. Math. Soc. **106** (1989), 821-828

MSC:
Primary 55P55; Secondary 03E35, 55N07

MathSciNet review:
961403

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Abstract: This paper concerns applications of set theory to the problem of calculating the strong homology of certain subsets of Euclidean spaces. We prove the set theoretic result that it is consistent that every almost coinciding family indexed by is trivial (e.g., the proper forcing axiom implies this). This result, combined with results of S. Mardešić and A. Prasalov, show that the statement "the -dimensional strong homology of (the discrete sum of countably many copies of the -dimensional Hawaiian earring) is trivial" is consistent with and independent of the usual axioms of set theory.

**[B]**James E. Baumgartner,*Applications of the proper forcing axiom*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 913–959. MR**776640****[vD]**Eric K. van Douwen,*The integers and topology*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR**776622****[LM]**Juriĭ Lisica and Sibe Mardešić,*Strong homology of inverse systems. III*, Topology Appl.**20**(1985), no. 1, 29–37. MR**798442**, 10.1016/0166-8641(85)90032-X**[M]**J. Milnor,*On axiomatic homology theory*, Pacific J. Math.**12**(1962), 337–341. MR**0159327****[MP]**S. Mardešić and A. V. Prasolov,*Strong homology is not additive*, Trans. Amer. Math. Soc.**307**(1988), no. 2, 725–744. MR**940224**, 10.1090/S0002-9947-1988-0940224-7

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0961403-5

Keywords:
Strong homology,
Hawaiian earring,
almost coinciding families,
continuum hypothesis,
bounding number,
dominating number,
-gaps in the sense of Hausdorff,
-scale

Article copyright:
© Copyright 1989
American Mathematical Society