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Igor Rivin
Igor Rivin
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In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another dimensiondirection there is a bound, and this is discussed in the nice paper below (on the first page):

Le Donne, Enrico; Rajala, Tapio, Assouad dimension, Nagata dimension, and uniformly close metric tangents, Indiana Univ. Math. J. 64, No. 1, 21-54 (2015). ZBL1321.54059.

In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another dimension there is a bound, and this is discussed in the nice paper below (on the first page):

Le Donne, Enrico; Rajala, Tapio, Assouad dimension, Nagata dimension, and uniformly close metric tangents, Indiana Univ. Math. J. 64, No. 1, 21-54 (2015). ZBL1321.54059.

In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another direction there is a bound, and this is discussed in the nice paper below (on the first page):

Le Donne, Enrico; Rajala, Tapio, Assouad dimension, Nagata dimension, and uniformly close metric tangents, Indiana Univ. Math. J. 64, No. 1, 21-54 (2015). ZBL1321.54059.

Source Link
Igor Rivin
Igor Rivin
  • 96.4k
  • 11
  • 153
  • 366

In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another dimension there is a bound, and this is discussed in the nice paper below (on the first page):

Le Donne, Enrico; Rajala, Tapio, Assouad dimension, Nagata dimension, and uniformly close metric tangents, Indiana Univ. Math. J. 64, No. 1, 21-54 (2015). ZBL1321.54059.