1,124 research outputs found
Compact maps and quasi-finite complexes
The simplest condition characterizing quasi-finite CW complexes is the
implication for all paracompact spaces .
Here are the main results of the paper:
Theorem: If is a family of pointed quasi-finite complexes,
then their wedge is quasi-finite.
Theorem: If and are quasi-finite countable complexes, then their
join is quasi-finite.
Theorem: For every quasi-finite CW complex there is a family
of countable CW complexes such that is quasi-finite and is equivalent, over the class of paracompact spaces,
to .
Theorem: Two quasi-finite CW complexes and are equivalent over the
class of paracompact spaces if and only if they are equivalent over the class
of compact metric spaces.
Quasi-finite CW complexes lead naturally to the concept of , where is a family of maps between CW complexes. We
generalize some well-known results of extension theory using that concept.Comment: 20 page
Dimension zero at all scales
We consider the notion of dimension in four categories: the category of
(unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and
the category of (unbounded) separable metric spaces and (metrically proper)
uniform maps. A unified treatment is given to the large scale dimension and the
small scale dimension. We show that in all categories a space has dimension
zero if and only if it is equivalent to an ultrametric space. Also,
0-dimensional spaces are characterized by means of retractions to subspaces.
There is a universal zero-dimensional space in all categories. In the Lipschitz
Category spaces of dimension zero are characterized by means of extensions of
maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is
coarsely equivalent to a direct sum of cyclic groups. We construct uncountably
many examples of coarsely inequivalent ultrametric spaces.Comment: 17 pages, To appear in Topology and its Application
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