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Is it possible to have a Hausdorff dimension less than the topological dimension?

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"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological dimension. Is there a class of shapes that have a Hausdorff dimension less than their topological dimension, or is this impossible? If there is such a shape, what are common examples of them? If this is impossible, why?

dimension-theory hausdorff-measure

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edited Mar 10 at 3:23

J. W. Tanner

3,2201320

asked Mar 10 at 3:15

tox123tox123

562721

$endgroup$

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  $begingroup$
  No, covering dimension is always $le$ Hausdorff dimension (Sznirelman's theorem).
  $endgroup$
  – Moishe Kohan
  Mar 10 at 3:30
  

  
  
  
  

add a comment | 

0

1

$begingroup$

"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological dimension. Is there a class of shapes that have a Hausdorff dimension less than their topological dimension, or is this impossible? If there is such a shape, what are common examples of them? If this is impossible, why?

dimension-theory hausdorff-measure

share|cite|improve this question

edited Mar 10 at 3:23

J. W. Tanner

3,2201320

asked Mar 10 at 3:15

tox123tox123

562721

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  No, covering dimension is always $le$ Hausdorff dimension (Sznirelman's theorem).
  $endgroup$
  – Moishe Kohan
  Mar 10 at 3:30
  

  
  
  
  

add a comment | 

$begingroup$

"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological dimension. Is there a class of shapes that have a Hausdorff dimension less than their topological dimension, or is this impossible? If there is such a shape, what are common examples of them? If this is impossible, why?

dimension-theory hausdorff-measure

share|cite|improve this question

edited Mar 10 at 3:23

J. W. Tanner

3,2201320

asked Mar 10 at 3:15

tox123tox123

562721

$endgroup$

share|cite|improve this question

edited Mar 10 at 3:23

J. W. Tanner

3,2201320

asked Mar 10 at 3:15

tox123tox123

562721

share|cite|improve this question

edited Mar 10 at 3:23

J. W. Tanner

3,2201320

asked Mar 10 at 3:15

tox123tox123

562721

edited Mar 10 at 3:23

J. W. Tanner

3,2201320

edited Mar 10 at 3:23

J. W. Tanner

3,2201320

asked Mar 10 at 3:15

tox123tox123

562721

asked Mar 10 at 3:15

tox123tox123

562721