Some extension theorems for continuous functions

Michael, Ernest A.

doi:10.2140/pjm.1953.3.789

Cited by 70 publications

(30 citation statements)

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“…Consequently, by a result of E. Michael [39], Y is an AE for all paracompact spaces. This implies that f −1 : f (A/G) → A/G → Y extends to a continuous map F : B → Y .…”

Section: Equivariant Lifting Of Closed Embeddingsmentioning

confidence: 84%

Proper actions of locally compact groups on equivariant absolute extensors

Antonyan

2009

Fund. Math.

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Abstract. Let G be a locally compact Hausdorff group. We study equivariant absolute (neighborhood) extensors (G-AE's and G-ANE's) in the category G-M of all proper G-spaces that are metrizable by a G-invariant metric. We first solve the linearization problem for proper group actions by proving that each X ∈ G-M admits an equivariant embedding in a Banach G-space L such that L\{0} is a proper G-space and L\{0} ∈ G-AE. This implies that in G-M the notions of G-A(N)E and G-A(N)R coincide. Our embedding result is applied to prove that if a G-space X is a G-ANE (resp., a G-AE) such that all the orbits in X are metrizable, then the orbit space X/G is an ANE (resp., an AE if, in addition, G is almost connected). Furthermore, we prove that if X ∈ G-M then for any closed embedding X/G → B in a metrizable space B, there exists a closed G-embedding X → Z (a lifting) in a G-space Z ∈ G-M such that Z/G is a neighborhood of X/G (resp., Z/G = B whenever G is almost connected). If a proper G-space X has metrizable orbits and a metrizable orbit space then it is metrizable (by a G-invariant metric).

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“…Consequently, by a result of E. Michael [39], Y is an AE for all paracompact spaces. This implies that f −1 : f (A/G) → A/G → Y extends to a continuous map F : B → Y .…”

Section: Equivariant Lifting Of Closed Embeddingsmentioning

confidence: 84%

Proper actions of locally compact groups on equivariant absolute extensors

Antonyan

2009

Fund. Math.

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“…Supposing A is a closed G% set in the (collectionwise) normal space (X, T), A is the zero set of some Remark. This note grew out of an attempt to verify an assertion appearing on p. 806 of [6] where it is asserted that simultaneous extenders from A to X will exist provided A is a metrizable closed subset of a paracompact space X (the hypothesis that A is also a G g was inadvertently omitted). …”

Section: Lutzer1 and H Martinmentioning

confidence: 99%

A note on the Dugundji extension theorem

Lutzer¹,

Martin²

1974

Proc. Amer. Math. Soc.

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ABSTRACT.We prove that if A is a closed, metrizable, Gg-subspace of a collectionwise normal space X then there is a linear transformation e: C(A) -* C(X) such that for each g £ C(A), e(g) extends g and the range of e(g) is contained in the closed convex hull of the range of g. topology M on X having M C T.The second major ingredient in our proof is a slight generalization of the original Dugundji extension theorem wherein we consider pseudo-metriz-

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“…Bearing in mind that point-finite refinements may be made precise [2, Theorem VIII 1.4], one may adapt Michael's proof of Theorem 2 in [5] to show spaces both N-metacompact and X-collectionwise normal are N-paracompact. Similarly, adapt McAuley's proof of Lemma 2 in [4] to handle the case where Z is semimetric.…”

Section: Proofmentioning

confidence: 99%