Contracting spaces of maps on the countable direct limit of a space

Author:
Richard E. Heisey

Journal:
Trans. Amer. Math. Soc. **193** (1974), 389-411

MSC:
Primary 54C35

DOI:
https://doi.org/10.1090/S0002-9947-1974-0367908-6

MathSciNet review:
0367908

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Abstract | References | Similar Articles | Additional Information

Abstract: We give conditions sufficient to imply the contractibility of the space of maps, with compact-open topology, on the countable direct limit of a space. Applying these conditions we obtain the following: Let *F* be the conjugate of a separable infinite-dimensional Banach space with bounded weak- topology, or the countable direct limit of the real line. Then there is a contraction of the space of maps on *F* which simultaneously contracts the subspaces of open maps, embeddings, closed embeddings, and homeomorphisms. Corollaries of our work are that any homeomorphism on *F, F* as above, is invertibly isotopic to the identity, and the general linear group of the countable direct limit of lines is contractible.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0367908-6

Keywords:
Direct limit,
function space,
topological vector space,
Banach space,
bounded weak- topology,
Hilbert cube

Article copyright:
© Copyright 1974
American Mathematical Society