Glasnik Matematicki, Vol. 45, No.2 (2010), 525-530.

ON INVERSE LIMITS OF COMPACT SPACES. CORRECTION OF A PROOF

Sibe Mardešić

Department of Mathematics, University of Zagreb, P.O.Box 335, 10 002 Zagreb, Croatia
e-mail: smardes@math.hr

Abstract.   For a compact Hausdorff space X and an ANR for metrizable spaces M, one considers the space MX of all mappings from X to M, endowed with the compact-open topology. Since a mapping f: X' → X induces a natural mapping Mf : MX → MX', an inverse system of compact Hausdorff spaces X determines a direct system M X of spaces as well as the corresponding direct system of singular homology groups Hn(M X;G). There is a natural isomorphism between the direct limit dir lim Hn(M X;G) and the singular homology group Hn(MX;G), where X= inv lim X. This continuity theorem, used by some authors, was published more than 50 years ago. Unfortunately, the author discovered a serious error in the proofs of two lemmas on which the result depended. The present paper gives new correct proofs of these lemmas.

2000 Mathematics Subject Classification.   54C35, 54B35, 54C55, 55N10.

Key words and phrases.   Homology of spaces of mappings, compact Hausdorff space, absolute neighborhood retract, absolute neighborhood extensor, direct system, inverse system.

Full text (PDF) (free access)

DOI: 10.3336/gm.45.2.17

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