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Register a new userEquivariant embedding of metrizable $G$-spaces in linear $G$-spaces
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Authors
Aasa Feragen
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Given a Lie group $G$ we study the class $M$ of proper metrizable $G$-spaces with metrizable orbit spaces, and show that any $G$-space $X in M$ admits a closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear $G$-space, such that $X$ has some $G$-neighborhood in $C$ which belongs to the class $M$. As corollaries we see that any $G$-ANE for $M$ has the $G$-homotopy type of some $G$-CW complex and that any $G$-ANR for $M$ is a $G$-ANE for $M$.
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We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.
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