Local dimension
of a normal topological space
The topological invariant , defined as follows. One says that
,
if for any point
there is a neighbourhood
for which the Lebesgue dimension of its closure satisfies the relation
. If
for some
, then the local dimension of
is finite, so one writes
and puts
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Always ; there are normal spaces
with
; in the class of paracompact spaces always
. If in the definition of local dimension the Lebesgue dimension
is replaced by the large inductive dimension
, then one obtains the definition of the local large inductive dimension
.
Comments
See [a1] for a construction of a space with and — as an application — a hereditarily normal space
with
yet
contains subspaces of arbitrary high dimension.
For the notions of the local dimension at a point of an analytic space, algebraic variety or scheme cf. Analytic space; Dimension of an associative ring; Analytic set, and Spectrum of a ring.
References
[a1] | E. Pol, R. Pol, "A hereditarily normal strongly zero-dimensional space containing subspaces of arbitrarily large dimension" Fund. Math. , 102 (1979) pp. 137–142 |
[a2] | R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50 |
Local dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_dimension&oldid=33942