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edited Dec 30, 2014 at 23:14

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In a metrizable topological space, Hausdorff dimension is always larger or equal than the topological (covering) dimension. See Theorem 6.3.10 in Edgar's book "Measure, topology and fractal geometry". In particular, for an $n$-dimensional manifold $M$, if $\rho$ is any metric compatible with the Euclidean topology, then $(M,\rho)$ has Hausdorff dimension at at least $d$$n$.

Because Hausdorff dimension is always at least the topological dimension, and both agree for "nice" spaces such as manifolds, Mandelbrot tentatively defined a fractal to be a metric space whose Hausdorff dimension is strictly larger than its similarity dimension, although this definition is not widely accepted today (the consensus is that you can't really define fractal).

Finally, for any (at least separable) metric space $X$, the topological dimension equals the infimum (which is in fact a minimum) of the Hausdorff dimensions of $(X,\rho)$ where $\rho$ varies among the metrics compatible with the topology of $X$. This is a classical result due to Edward Marczewski.

In a metrizable topological space, Hausdorff dimension is always larger or equal than the topological (covering) dimension. See Theorem 6.3.10 in Edgar's book "Measure, topology and fractal geometry". In particular, for an $n$-dimensional manifold $M$, if $\rho$ is any metric compatible with the Euclidean topology, then $(M,\rho)$ has Hausdorff dimension at least $d$.

Because Hausdorff dimension is always at least the topological dimension, and both agree for "nice" spaces such as manifolds, Mandelbrot tentatively defined a fractal to be a metric space whose Hausdorff dimension is strictly larger than its similarity dimension, although this definition is not widely accepted today (the consensus is that you can't really define fractal).

Finally, for any (at least separable) metric space $X$, the topological dimension equals the infimum (which is in fact a minimum) of the Hausdorff dimensions of $(X,\rho)$ where $\rho$ varies among the metrics compatible with the topology of $X$. This is a classical result due to Edward Marczewski.

In a metrizable topological space, Hausdorff dimension is always larger or equal than the topological (covering) dimension. See Theorem 6.3.10 in Edgar's book "Measure, topology and fractal geometry". In particular, for an $n$-dimensional manifold $M$, if $\rho$ is any metric compatible with the Euclidean topology, then $(M,\rho)$ has Hausdorff dimension at least $n$.

Because Hausdorff dimension is always at least the topological dimension, and both agree for "nice" spaces such as manifolds, Mandelbrot tentatively defined a fractal to be a metric space whose Hausdorff dimension is strictly larger than its similarity dimension, although this definition is not widely accepted today (the consensus is that you can't really define fractal).

Finally, for any (at least separable) metric space $X$, the topological dimension equals the infimum (which is in fact a minimum) of the Hausdorff dimensions of $(X,\rho)$ where $\rho$ varies among the metrics compatible with the topology of $X$. This is a classical result due to Edward Marczewski.

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created Dec 30, 2014 at 14:34

4.7k
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33

In a metrizable topological space, Hausdorff dimension is always larger or equal than the topological (covering) dimension. See Theorem 6.3.10 in Edgar's book "Measure, topology and fractal geometry". In particular, for an $n$-dimensional manifold $M$, if $\rho$ is any metric compatible with the Euclidean topology, then $(M,\rho)$ has Hausdorff dimension at least $d$.

Because Hausdorff dimension is always at least the topological dimension, and both agree for "nice" spaces such as manifolds, Mandelbrot tentatively defined a fractal to be a metric space whose Hausdorff dimension is strictly larger than its similarity dimension, although this definition is not widely accepted today (the consensus is that you can't really define fractal).

Finally, for any (at least separable) metric space $X$, the topological dimension equals the infimum (which is in fact a minimum) of the Hausdorff dimensions of $(X,\rho)$ where $\rho$ varies among the metrics compatible with the topology of $X$. This is a classical result due to Edward Marczewski.