Abstract:Let $G$ be a compact group. The existence of certain $G$-homotopy dense subsets in a metrizable $G$-space $X$ plays a fundamental role, as it is equivalent to $X$ being a $G$-ANR. From this perspective, the present paper develops several applications of this class of $G$-subsets.
In particular, we prove that for a compact $G$-space $X$ and a metric space $Y$, the mapping space $C(X,Y)$ is a $G$-UA(N)R if and only if $Y$ is a UA(N)R in the sense of Michael. This result is significant because it enables the construction of examples of Lawson metric $G$-semilattices for which the property of being a $G$-UANR is equivalent to uniform local path-connectedness. Moreover, we show that this equivalence holds for every Lawson metric $G$-semilattice whenever $G$ is finite.
Finally, we analyze the behavior of $G$-homotopy dense subsets when the ambient space is a $G$-A(N)R, thereby introducing the notion of a $G$-A(N)R-pair.
| Subjects: | General Topology (math.GN) |
| Cite as: | arXiv:2605.24141 [math.GN] |
| (or arXiv:2605.24141v1 [math.GN] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24141 arXiv-issued DOI via DataCite (pending registration) |
From: Sergey Antonyan Prof. [view email]
[v1]
Fri, 22 May 2026 19:03:36 UTC (18 KB)
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