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A method for characterizing the dimension of a compactum lying in a Euclidean space  $  \mathbf R  ^ {n} $
in terms of metric properties of the complementary space. The essential measure of a cycle  $  z $
in a compactum  $  \Phi \subset  \mathbf R  ^ {n} $
is taken to be the least upper bound of those  $  \epsilon > 0 $
for which it is possible to select a compact support  $  \Phi _ {1} \subseteq \Phi $
of the cycle  $  z $
such that the cycle is not homologous to zero in  $  O ( \Phi _ {1} , \epsilon ) $. 
The  $  p $-
dimensional homological diameter  $  \alpha _  \Gamma   ^ {p} z $
of a cycle  $  z $
in an open set  $  \Gamma = \mathbf R  ^ {n} \setminus  \Phi $
is the greatest lower bound of the  $  p $-
dimensional diameters of the bodies of all cycles that are homologous in  $  \Gamma $
to  $  z $. 
Here, the  $  p $-
dimensional diameter  $  \alpha  ^ {p} X $
of a compactum  $  X \subset  \mathbf R  ^ {n} $
is the greatest lower bound of those  $  \epsilon > 0 $
for which there exists a continuous  $  \epsilon $-
shift of  $  X $
in a  $  p $-
dimensional compactum (and thus in a polyhedron).

Any  $  ( n - 1 ) $-
dimensional cycle of the open set  $  \Gamma = \mathbf R  ^ {n} \setminus  \Phi $
which is linked with each point of the compactum  $  \Phi $
is said to be a pocket around  $  \Phi $.

The pocket theorem. Let  $  r =  \mathop{\rm dim}  \Phi \leq  n - 1 $. 
Then there exists an  $  \alpha > 0 $
such that any pocket around  $  \Phi $
has  $  ( r - 1) $-
dimensional homological diameter larger than  $  \alpha $, 
while the  $  r $-
dimensional homological diameter of any cycle in  $  \Gamma $
is zero. Pockets around  $  \Phi $
with arbitrary small essential measure always exist in this situation. On the other hand, if  $   \mathop{\rm dim}  \Phi = n $, 
then there exists an  $  \alpha > 0 $
such that for any pocket  $  z  ^ {n-1} $
around  $  \Phi $
the inequality  $  \mu z  ^ {n-1} > \alpha $
is true (here,  $  \alpha _  \Gamma   ^ {n-2} z  ^ {n-1} > 0 $
and  $  \alpha _  \Gamma   ^ {n-1} z  ^ {n-1} = 0 $
for any pocket  $  z  ^ {n-1} $).

The pocket theorem may be further strengthened using the concept of a zone around a compactum.

The zone theorem. Let  $  \Phi \subset  \mathbf R  ^ {n} $
be a compactum of dimension  $  r $. 
There exists a  $  \gamma > 0 $
such that for any  $  k = 1 \dots r + 1 $
and any  $  \epsilon > 0 $
there exists in  $  \Gamma = \mathbf R  ^ {n} \setminus  \Phi $
an  $  ( n - k) $-
dimensional cycle  $  v $(
a zone of dimension  $  n - k $
around  $  \Phi $), 
for  $  k> 1 $
bounding in  $  \Gamma $, 
for which  $  \beta  ^ {r-k+ 1} v < \epsilon $, 
$  \tau v < \epsilon $. 
Furthermore, for any cycle  $  w $
homologous to  $  v $
in the  $  \gamma $-
neighbourhood of the latter with respect to  $  \Gamma $
the inequality  $  \beta  ^ {r-n+ 1} w > \gamma $
is valid; for any chain  $  x $
bounded by the cycle  $  v $
in  $  \Gamma $
one has  $  \beta  ^ {r-n+ 1} x > \gamma $.

On the other hand, if  $  s > r $
and if  $  k = 1 \dots s + 1 $, 
then, for any  $  \gamma > 0 $, 
any  $  ( n - k) $-
dimensional cycle  $  z $
in  $  \Gamma $
for which  $  \tau z < \gamma $
is homologous in its  $  \gamma $-
neighbourhood (with respect to  $  \Gamma $) 
to some cycle  $  z  ^  \prime  $
with  $  \beta  ^ {s-k} z  ^  \prime  $
arbitrary small. Furthermore, if  $  s > r $
and if  $  k = 2 \dots s + 1 $, 
then for any  $  \gamma > 0 $
any  $  ( n - k ) $-
dimensional cycle  $  z $, 
bounding in  $  \Gamma $, 
for which  $  \beta  ^ {s-k+ 1} z < \gamma $(
and  $  \tau z < \gamma $
if  $  s = n - 1 $) 
bounds in  $  \Gamma $
a chain  $  x $
with  $  \beta  ^ {s-n+ 1} x < \gamma $. 
Here  $  \beta  ^ {p} x $, 
$  p \geq  0 $, 
is the greatest lower bound of those  $  \epsilon > 0 $
for which there exists an  $  \epsilon $-
shift of the vertices of the chain  $  x $
by means of which  $  x $
becomes degenerate up to dimension  $  p $; 
$  \tau x $
is the greatest lower bound of those  $  \epsilon > 0 $
for which there exists an  $  \epsilon $-
shift of the vertices of  $  x $
converting  $  x $
to a zero chain.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "An introduction to homological dimension theory and general combinatorial topology" , Moscow  (1975)  (In Russian)</TD></TR></table>

====Comments====
An $\epsilon$-shift of a subspace  $  X $ contained in some Euclidean space  $  \mathbf R  ^ {m} $
is a mapping  $  f:  X \rightarrow \mathbf R  ^ {m} $
such that the distance of  $  x $
to  $  f( x) $
is less than  $  \epsilon $
for all  $  x \in X $.