Extending homeomorphisms and applications to metric linear spaces without completeness

Dobrowolski, Tadeusz

doi:10.1090/S0002-9947-1989-0930078-8

Extending homeomorphisms and applications to metric linear spaces without completeness
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by Tadeusz Dobrowolski

Trans. Amer. Math. Soc. 313 (1989), 753-784

DOI: https://doi.org/10.1090/S0002-9947-1989-0930078-8

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Abstract:

A method of extending homeomorphisms between compacta metric spaces is presented. The main application is that homeomorphisms between compacta of an infinite-dimensional locally convex metric linear space extend to the whole space. A lemma used in the proof of this fact together with the method of absorbing sets is employed to show that every $\sigma$ -compact normed linear space is homeomorphic to a dense linear subspace of a Hilbert space. A discussion of the relative topological equivalence of absorbing sets in noncomplete spaces is included. The paper is concluded with some controlled versions of an isotopy extension theorem.

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