Infinite dimension of proper $G$-spaces
Volume 178 / 2025
Abstract
We continue the study of the covering dimension of proper $G$-spaces initiated in [Topology Appl. 301 (2021), art. 107491]. For a Lie (or in some cases, a finite-dimensional locally compact) group $G$, we consider, in the class $G$-$\mathcal {M}$ of proper $G$-spaces that admit an invariant metric compatible with their topology, three kinds of infinite dimension: countable, strongly countable and locally finite dimension. We prove that $X\in G$-$\mathcal {M}$ is a (strongly) countable-dimensional (respectively locally finite-dimensional) $G$-space if and only if $X/G$ is a (strongly) countable-dimensional (respectively locally finite-dimensional) space. As a by-product we obtain a factorization theorem with respect to the weight and the strongly countable (respectively, locally finite) dimension and we prove that for each infinite cardinal $\tau $, the class of strongly countable-dimensional (respectively locally finite-dimensional) proper $G$-spaces of weight less than or equal to $\tau $ has a universal element.
