Given some space $X$ with non-finite Assouad-Nagata dimension. Is it possible for a subset $Y \subset X$ with finite Assouad-Nagata dimension to exist such that $Y$ is maximal in the sense that if any other point $x_0$ in $X$ would be added to $Y$ then the Assouad-Nagata dimension of $X \cup \{x_0\}$ would become infinite?
2026-03-28 00:48:36.1774658916
Maximal subset with finite Assouad-Nagata dimension
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No. The Assouad-Nagata dimension is stable under finite union, meaning that $\dim_{AN} (X\cup Y) = \max\{\dim_{AN}X, \dim_{AN}Y)$. Since a point has AN-dimension $0$, adding it to any set does not increase its AN-dimension.